diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index 09e1853429d9f..d8c4f633d2d0d 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -287,6 +287,7 @@ Datetimelike
 - Bug in :meth:`Timestamp.round`, :meth:`Timestamp.floor`, :meth:`Timestamp.ceil` for values near the implementation bounds of :class:`Timestamp` (:issue:`39244`)
 - Bug in :meth:`Timedelta.round`, :meth:`Timedelta.floor`, :meth:`Timedelta.ceil` for values near the implementation bounds of :class:`Timedelta` (:issue:`38964`)
 - Bug in :func:`date_range` incorrectly creating :class:`DatetimeIndex` containing ``NaT`` instead of raising ``OutOfBoundsDatetime`` in corner cases (:issue:`24124`)
+- Bug in :func:`infer_freq` incorrectly fails to infer 'H' frequency of :class:`DatetimeIndex` if the latter has a timezone and crosses DST boundaries (:issue:`39556`)
 
 Timedelta
 ^^^^^^^^^
diff --git a/pandas/tests/tseries/frequencies/test_inference.py b/pandas/tests/tseries/frequencies/test_inference.py
index 95edd038dab9b..892be119093e5 100644
--- a/pandas/tests/tseries/frequencies/test_inference.py
+++ b/pandas/tests/tseries/frequencies/test_inference.py
@@ -267,7 +267,7 @@ def test_infer_freq_tz(tz_naive_fixture, expected, dates):
     ],
 )
 @pytest.mark.parametrize(
-    "freq", ["3H", "10T", "3601S", "3600001L", "3600000001U", "3600000000001N"]
+    "freq", ["H", "3H", "10T", "3601S", "3600001L", "3600000001U", "3600000000001N"]
 )
 def test_infer_freq_tz_transition(tz_naive_fixture, date_pair, freq):
     # see gh-8772
diff --git a/pandas/tseries/frequencies.py b/pandas/tseries/frequencies.py
index 0d5598fcaf890..2257ce47d8936 100644
--- a/pandas/tseries/frequencies.py
+++ b/pandas/tseries/frequencies.py
@@ -240,16 +240,17 @@ def get_freq(self) -> Optional[str]:
             return None
 
         delta = self.deltas[0]
-        if _is_multiple(delta, _ONE_DAY):
+        if delta and _is_multiple(delta, _ONE_DAY):
             return self._infer_daily_rule()
 
         # Business hourly, maybe. 17: one day / 65: one weekend
         if self.hour_deltas in ([1, 17], [1, 65], [1, 17, 65]):
             return "BH"
+
         # Possibly intraday frequency.  Here we use the
         # original .asi8 values as the modified values
         # will not work around DST transitions.  See #8772
-        elif not self.is_unique_asi8:
+        if not self.is_unique_asi8:
             return None
 
         delta = self.deltas_asi8[0]