BUG: Fix dt64[non_nano] + some_offsets incorrectly rounding

pandas/core/arrays/datetimes.py

@@ -817,7 +817,16 @@ def _add_offset(self, offset: BaseOffset) -> Self:
             result = type(self)._from_sequence(res_values, dtype=self.dtype)
 
         else:
-            result = type(self)._simple_new(res_values, dtype=res_values.dtype)
+            units = ["ns", "us", "ms", "s", "m", "h", "D"]
+            res_unit = self.unit
+            if hasattr(offset, "offset"):
+                offset_unit = Timedelta(offset.offset).unit
+                idx_self = units.index(self.unit)
+                idx_offset = units.index(offset_unit)
+                res_unit = units[min(idx_self, idx_offset)]
+            dtype = tz_to_dtype(self.tz, unit=res_unit)
+            result = type(self)._simple_new(res_values, dtype=dtype)
+
             if offset.normalize:
                 result = result.normalize()
                 result._freq = None

pandas/tests/arrays/test_datetimes.py

@@ -844,3 +844,20 @@ def test_factorize_sort_without_freq():
     tda = dta - dta[0]
     with pytest.raises(NotImplementedError, match=msg):
         tda.factorize(sort=True)
+
+
+def test_dt64_non_nano_offset_no_rounding():
+    # GH#56586
+    dti = pd.date_range("2016-01-01", periods=3, unit="s")
+    offset = pd.offsets.CustomBusinessDay(offset=pd.Timedelta("1ms"))
+    result = dti + offset
+
+    assert result.dtype == np.dtype("datetime64[ms]")
+    expected = pd.DatetimeIndex(
+        [
+            pd.Timestamp("2016-01-02 00:00:00.001"),
+            pd.Timestamp("2016-01-03 00:00:00.001"),
+            pd.Timestamp("2016-01-04 00:00:00.001"),
+        ]
+    )
+    assert all(result == expected)