diff --git a/src/toil/jobStores/fileJobStore.py b/src/toil/jobStores/fileJobStore.py
index 0ae5c8816b..fa6df1d325 100644
--- a/src/toil/jobStores/fileJobStore.py
+++ b/src/toil/jobStores/fileJobStore.py
@@ -66,6 +66,11 @@ class FileJobStore(AbstractJobStore):
     # 10Mb RAM chunks when reading/writing files
     BUFFER_SIZE = 10485760  # 10Mb
 
+    # When a log file is still being written, what will its name end with?
+    LOG_TEMP_SUFFIX = ".new"
+    # All log files start with this prefix
+    LOG_PREFIX = "stats"
+
     def default_caching(self) -> bool:
         """
         Jobstore's preference as to whether it likes caching or doesn't care about it.
@@ -90,6 +95,9 @@ def __init__(self, path: str, fanOut: int = 1000) -> None:
         self.jobsDir = os.path.join(self.jobStoreDir, "jobs")
         # Directory where stats files go
         self.statsDir = os.path.join(self.jobStoreDir, "stats")
+        # Which has subdirectories for new and seen stats files
+        self.stats_inbox = os.path.join(self.statsDir, "inbox")
+        self.stats_archive = os.path.join(self.statsDir, "archive")
         # Directory where non-job-associated files for the file store go
         self.filesDir = os.path.join(self.jobStoreDir, "files/no-job")
         # Directory where job-associated files for the file store go.
@@ -118,6 +126,8 @@ def initialize(self, config):
                 raise
         os.makedirs(self.jobsDir, exist_ok=True)
         os.makedirs(self.statsDir, exist_ok=True)
+        os.makedirs(self.stats_inbox, exist_ok=True)
+        os.makedirs(self.stats_archive, exist_ok=True)
         os.makedirs(self.filesDir, exist_ok=True)
         os.makedirs(self.jobFilesDir, exist_ok=True)
         os.makedirs(self.sharedFilesDir, exist_ok=True)
@@ -836,29 +846,66 @@ def list_all_file_names(self, for_job: Optional[str] = None) -> Iterable[str]:
 
     def write_logs(self, msg):
         # Temporary files are placed in the stats directory tree
-        tempStatsFileName = "stats" + str(uuid.uuid4().hex) + ".new"
-        tempStatsFile = os.path.join(self._get_arbitrary_stats_dir(), tempStatsFileName)
+        tempStatsFileName = self.LOG_PREFIX + str(uuid.uuid4().hex) + self.LOG_TEMP_SUFFIX
+        tempStatsFile = os.path.join(self._get_arbitrary_stats_inbox_dir(), tempStatsFileName)
         writeFormat = "w" if isinstance(msg, str) else "wb"
         with open(tempStatsFile, writeFormat) as f:
             f.write(msg)
-        os.rename(tempStatsFile, tempStatsFile[:-4])  # This operation is atomic
+        os.rename(tempStatsFile, tempStatsFile[:-len(self.LOG_TEMP_SUFFIX)])  # This operation is atomic
 
     def read_logs(self, callback, read_all=False):
-        numberOfFilesProcessed = 0
-        for tempDir in self._stats_directories():
-            for tempFile in os.listdir(tempDir):
-                if tempFile.startswith("stats"):
-                    absTempFile = os.path.join(tempDir, tempFile)
-                    if os.path.isfile(absTempFile):
-                        if read_all or not tempFile.endswith(".new"):
-                            with open(absTempFile, "rb") as fH:
-                                callback(fH)
-                            numberOfFilesProcessed += 1
-                            newName = tempFile.rsplit(".", 1)[0] + ".new"
-                            newAbsTempFile = os.path.join(tempDir, newName)
+        files_processed = 0
+
+        # Holds pairs of a function to call to get directories to look at, and
+        # a flag for whether to archive the files found.
+        queries = []
+        if read_all:
+            # If looking at all logs, check the archive
+            queries.append((self._stats_archive_directories, False))
+        # Always check the inbox and archive from it. But do it after checking
+        # the archive to avoid duplicates in the same pass.
+        queries.append((self._stats_inbox_directories, True))
+
+        for to_call, should_archive in queries:
+            for log_dir in to_call():
+                for log_file in os.listdir(log_dir):
+                    if not log_file.startswith(self.LOG_PREFIX):
+                        # Skip anything not a log file (like the other spray
+                        # directories)
+                        continue
+                    if log_file.endswith(self.LOG_TEMP_SUFFIX):
+                        # Skip partially-written files, always.
+                        continue
+
+                    abs_log_file = os.path.join(log_dir, log_file)
+                    if not os.path.isfile(abs_log_file):
+                        # This can't be a log file.
+                        continue
+                    try:
+                        opened_file = open(abs_log_file, "rb")
+                    except FileNotFoundError:
+                        # File disappeared before we could open it.
+                        # Maybe someone else is reading logs?
+                        continue
+                    with opened_file as f:
+                        callback(f)
+                    files_processed += 1
+
+                    if should_archive:
+                        # We need to move the stats file to the archive.
+                        # Since we have UUID stats file names we don't need
+                        # to worry about collisions when it gets there.
+                        new_dir = self._get_arbitrary_stats_archive_dir()
+                        new_abs_log_file = os.path.join(new_dir, log_file)
+                        try:
                             # Mark this item as read
-                            os.rename(absTempFile, newAbsTempFile)
-        return numberOfFilesProcessed
+                            os.rename(abs_log_file, new_abs_log_file)
+                        except FileNotFoundError:
+                            # File we wanted to archive disappeared.
+                            # Maybe someone else is reading logs?
+                            # TODO: Raise ConcurrentFileModificationException?
+                            continue
+        return files_processed
 
     ##########################################
     # Private methods
@@ -1010,9 +1057,10 @@ def _get_arbitrary_jobs_dir_for_name(self, jobNameSlug):
                 os.path.join(self.jobsDir, self.JOB_NAME_DIR_PREFIX + jobNameSlug)
             )
 
-    def _get_arbitrary_stats_dir(self):
+    def _get_arbitrary_stats_inbox_dir(self):
         """
-        Gets a temporary directory in a multi-level hierarchy in self.statsDir.
+        Gets a temporary directory in a multi-level hierarchy in
+        self.stats_inbox, where stats files not yet seen by the leader live.
         The directory is not unique and may already have other stats files in it.
 
         :rtype : string, path to temporary directory in which to place files/directories.
@@ -1020,7 +1068,20 @@ def _get_arbitrary_stats_dir(self):
 
         """
 
-        return self._get_dynamic_spray_dir(self.statsDir)
+        return self._get_dynamic_spray_dir(self.stats_inbox)
+
+    def _get_arbitrary_stats_archive_dir(self):
+        """
+        Gets a temporary directory in a multi-level hierarchy in
+        self.stats_archive, where stats files already seen by the leader live.
+        The directory is not unique and may already have other stats files in it.
+
+        :rtype : string, path to temporary directory in which to place files/directories.
+
+
+        """
+
+        return self._get_dynamic_spray_dir(self.stats_archive)
 
     def _get_arbitrary_files_dir(self):
         """
@@ -1156,14 +1217,23 @@ def _job_directories(self):
                     os.path.join(jobHoldingDir, jobNameDir)
                 )
 
-    def _stats_directories(self):
+    def _stats_inbox_directories(self):
         """
-        :rtype : an iterator to the temporary directories containing stats
-                 files. They may also contain directories containing more
-                 stats files.
+        :returns: an iterator to the temporary directories containing new stats
+            files. They may also contain directories containing more stats
+            files.
+        """
+
+        return self._walk_dynamic_spray_dir(self.stats_inbox)
+
+    def _stats_archive_directories(self):
+        """
+        :returns: an iterator to the temporary directories containing
+            previously observed stats files. They may also contain directories
+            containing more stats files.
         """
 
-        return self._walk_dynamic_spray_dir(self.statsDir)
+        return self._walk_dynamic_spray_dir(self.stats_archive)
 
     def _get_unique_file_path(self, fileName, jobStoreID=None, cleanup=False):
         """