diff --git a/.gitignore b/.gitignore
index dc6244855..da9a31ab5 100644
--- a/.gitignore
+++ b/.gitignore
@@ -48,3 +48,6 @@ tests/data/common_wheels/
 
 # Mac
 .DS_Store
+
+# Profiling related artifacts
+*.prof
diff --git a/news/9748.feature.rst b/news/9748.feature.rst
new file mode 100644
index 000000000..cb4a1cded
--- /dev/null
+++ b/news/9748.feature.rst
@@ -0,0 +1 @@
+Improve performance when picking the best file from indexes during `pip install`.
diff --git a/src/pip/_internal/index/package_finder.py b/src/pip/_internal/index/package_finder.py
index 4cc9ffd4d..a6423cce1 100644
--- a/src/pip/_internal/index/package_finder.py
+++ b/src/pip/_internal/index/package_finder.py
@@ -434,6 +434,12 @@ class CandidateEvaluator:
         self._project_name = project_name
         self._specifier = specifier
         self._supported_tags = supported_tags
+        # Since the index of the tag in the _supported_tags list is used
+        # as a priority, precompute a map from tag to index/priority to be
+        # used in wheel.find_most_preferred_tag.
+        self._wheel_tag_preferences = {
+            tag: idx for idx, tag in enumerate(supported_tags)
+        }
 
     def get_applicable_candidates(
         self,
@@ -512,14 +518,17 @@ class CandidateEvaluator:
         if link.is_wheel:
             # can raise InvalidWheelFilename
             wheel = Wheel(link.filename)
-            if not wheel.supported(valid_tags):
+            try:
+                pri = -(wheel.find_most_preferred_tag(
+                    valid_tags, self._wheel_tag_preferences
+                ))
+            except ValueError:
                 raise UnsupportedWheel(
                     "{} is not a supported wheel for this platform. It "
                     "can't be sorted.".format(wheel.filename)
                 )
             if self._prefer_binary:
                 binary_preference = 1
-            pri = -(wheel.support_index_min(valid_tags))
             if wheel.build_tag is not None:
                 match = re.match(r'^(\d+)(.*)$', wheel.build_tag)
                 build_tag_groups = match.groups()
diff --git a/src/pip/_internal/models/wheel.py b/src/pip/_internal/models/wheel.py
index 708bff330..c206d13cb 100644
--- a/src/pip/_internal/models/wheel.py
+++ b/src/pip/_internal/models/wheel.py
@@ -2,7 +2,7 @@
 name that have meaning.
 """
 import re
-from typing import List
+from typing import Dict, List
 
 from pip._vendor.packaging.tags import Tag
 
@@ -66,8 +66,26 @@ class Wheel:
         """
         return min(tags.index(tag) for tag in self.file_tags if tag in tags)
 
+    def find_most_preferred_tag(self, tags, tag_to_priority):
+        # type: (List[Tag], Dict[Tag, int]) -> int
+        """Return the priority of the most preferred tag that one of the wheel's file
+        tag combinations acheives in the given list of supported tags using the given
+        tag_to_priority mapping, where lower priorities are more-preferred.
+
+        This is used in place of support_index_min in some cases in order to avoid
+        an expensive linear scan of a large list of tags.
+
+        :param tags: the PEP 425 tags to check the wheel against.
+        :param tag_to_priority: a mapping from tag to priority of that tag, where
+            lower is more preferred.
+
+        :raises ValueError: If none of the wheel's file tags match one of
+            the supported tags.
+        """
+        return min(tag_to_priority[tag] for tag in self.file_tags if tag in tag_to_priority)
+
     def supported(self, tags):
-        # type: (List[Tag]) -> bool
+        # type: (Iterable[Tag]) -> bool
         """Return whether the wheel is compatible with one of the given tags.
 
         :param tags: the PEP 425 tags to check the wheel against.