diff --git a/CHANGELOG.rst b/CHANGELOG.rst
index 11d692483..7accb82ac 100644
--- a/CHANGELOG.rst
+++ b/CHANGELOG.rst
@@ -251,6 +251,7 @@ Completions
 - Lots of improvements to completions.
 - Improvements to the manpage completion generator (:issue:`7086`).
 - Significant performance improvements to completion of the available commands (:issue:`7153`).
+- ``__fish_complete_suffix`` completes any file but sorts files with matching suffix first (:issue:`7040`). Previously, it only completed files with matching suffix.
 
 Deprecations and removed features
 ---------------------------------
diff --git a/doc_src/completions.rst b/doc_src/completions.rst
index b4624780d..4731af5ef 100644
--- a/doc_src/completions.rst
+++ b/doc_src/completions.rst
@@ -83,7 +83,7 @@ Functions beginning with the string ``__fish_print_`` print a newline separated
 
 - ``__fish_complete_pids`` prints a list of all processes IDs with the command name as description.
 
-- ``__fish_complete_suffix SUFFIX`` performs file completion allowing only files ending in SUFFIX, with an optional description.
+- ``__fish_complete_suffix SUFFIX`` performs file completion but sorts files ending in SUFFIX first. This is useful in conjunction with ``complete --keep-order``.
 
 - ``__fish_complete_users`` prints a list of all users with their full name as description.
 
diff --git a/share/functions/__fish_complete_suffix.fish b/share/functions/__fish_complete_suffix.fish
index 5d0402dd8..37bef4648 100644
--- a/share/functions/__fish_complete_suffix.fish
+++ b/share/functions/__fish_complete_suffix.fish
@@ -1,10 +1,11 @@
 #
-# Find files that complete $argv[1], has the suffix $argv[2], and
-# output them as completions with the optional description $argv[3] Both
-# $argv[1] and $argv[3] are optional, if only one is specified, it is
-# assumed to be the argument to complete. If $argv[4] is present, it is
-# treated as a prefix for the path, i.e. in lieu of $PWD.
-#
+# Find files that complete $argv[1], has the suffix $argv[2], and output them
+# as completions with the optional description $argv[3].  Then, also output
+# completions for the files that don't have the suffix, so you want to use
+# "complete -k" on the output.  Both $argv[1] and $argv[3] are optional,
+# if only one is specified, it is assumed to be the argument to complete. If
+# $argv[4] is present, it is treated as a prefix for the path, i.e. in lieu
+# of $PWD.
 
 function __fish_complete_suffix -d "Complete using files"
 
@@ -48,17 +49,21 @@ function __fish_complete_suffix -d "Complete using files"
             end
     end
 
-    # Simple and common case: no prefix, just complete normally and filter out unwanted suffixes.
+    # Simple and common case: no prefix, just complete normally and sort matching files first.
     if test -z $prefix
         set -l suffix (string escape --style=regex -- $suff)
         # Use normal file completions.
-        set files (complete -C "__fish_command_without_completions $comp" | string match -r "^.*(?:$suffix|/)\$")
+        set files (complete -C "__fish_command_without_completions $comp")
+        set -l files_with_suffix (string match -r -- "^.*$suffix\$" $files)
+        set -l directories (string match -r -- '^.*/$' $files)
+        set files $files_with_suffix $directories $files
     else
         # Strip leading ./ as it confuses the detection of base and suffix
         # It is conditionally re-added below.
         set base $prefix(string replace -r '^("\')?\\./' '' -- $comp | string trim -c '\'"') # " make emacs syntax highlighting happy
 
         set -l all
+        set -l files_with_suffix
         set -l dirs
         # If $comp is "./ma" and the file is "main.py", we'll catch that case here,
         # but complete.cpp will not consider it a match, so we have to output the
@@ -67,7 +72,7 @@ function __fish_complete_suffix -d "Complete using files"
         # Also do directory completion, since there might be files with the correct
         # suffix in a subdirectory.
         set all $base*
-        set all (string match -r -- ".*"(string escape --style=regex -- $suff) $all)
+        set files_with_suffix (string match -r -- ".*"(string escape --style=regex -- $suff) $all)
         if not string match -qr '/$' -- $suff
             set dirs $base*/
 
@@ -82,7 +87,7 @@ function __fish_complete_suffix -d "Complete using files"
             end
         end
 
-        set files $all $dirs
+        set files $files_with_suffix $dirs $all
         if string match -qr '^\\./' -- $comp
             set files ./$files
         else