diff --git a/pandas/_libs/index.pyx b/pandas/_libs/index.pyx
index 1eca83f394987..f540dc99e91f3 100644
--- a/pandas/_libs/index.pyx
+++ b/pandas/_libs/index.pyx
@@ -157,19 +157,18 @@ cdef class IndexEngine:
 
         try:
             return self.mapping.get_item(val)
-        except (TypeError, ValueError, OverflowError):
+        except OverflowError as err:
             # GH#41775 OverflowError e.g. if we are uint64 and val is -1
-            raise KeyError(val)
+            #  or if we are int64 and value is np.iinfo(np.int64).max+1
+            #  (the uint64 with -1 case should actually be excluded by _check_type)
+            raise KeyError(val) from err
 
     cdef Py_ssize_t _searchsorted_left(self, val) except? -1:
         """
         See ObjectEngine._searchsorted_left.__doc__.
         """
-        try:
-            loc = self.values.searchsorted(val, side="left")
-        except TypeError as err:
-            # GH#35788 e.g. val=None with float64 values
-            raise KeyError(val)
+        # Caller is responsible for ensuring _check_type has already been called
+        loc = self.values.searchsorted(val, side="left")
         return loc
 
     cdef inline _get_loc_duplicates(self, object val):
@@ -184,6 +183,7 @@ cdef class IndexEngine:
                 right = values.searchsorted(val, side='right')
             except TypeError:
                 # e.g. GH#29189 get_loc(None) with a Float64Index
+                #  2021-09-29 Now only reached for object-dtype
                 raise KeyError(val)
 
             diff = right - left
diff --git a/pandas/_libs/index_class_helper.pxi.in b/pandas/_libs/index_class_helper.pxi.in
index 1b8e43ac24fa4..7a2bbec96e413 100644
--- a/pandas/_libs/index_class_helper.pxi.in
+++ b/pandas/_libs/index_class_helper.pxi.in
@@ -36,6 +36,11 @@ cdef class {{name}}Engine(IndexEngine):
     {{if name not in {'Float64', 'Float32'} }}
         if not util.is_integer_object(val):
             raise KeyError(val)
+        {{if name.startswith("U")}}
+        if val < 0:
+            # cannot have negative values with unsigned int dtype
+            raise KeyError(val)
+        {{endif}}
     {{else}}
         if not util.is_integer_object(val) and not util.is_float_object(val):
             # in particular catch bool and avoid casting True -> 1.0
diff --git a/pandas/tests/indexes/numeric/test_indexing.py b/pandas/tests/indexes/numeric/test_indexing.py
index dae8a3340bcd2..4621cbcb9d462 100644
--- a/pandas/tests/indexes/numeric/test_indexing.py
+++ b/pandas/tests/indexes/numeric/test_indexing.py
@@ -160,6 +160,17 @@ def test_get_loc_numericindex_none_raises(self, dtype):
         with pytest.raises(KeyError, match="None"):
             idx.get_loc(None)
 
+    def test_get_loc_overflows(self):
+        # unique but non-monotonic goes through IndexEngine.mapping.get_item
+        idx = Index([0, 2, 1])
+
+        val = np.iinfo(np.int64).max + 1
+
+        with pytest.raises(KeyError, match=str(val)):
+            idx.get_loc(val)
+        with pytest.raises(KeyError, match=str(val)):
+            idx._engine.get_loc(val)
+
 
 class TestGetIndexer:
     def test_get_indexer(self):