diff --git a/pandas/_libs/tslibs/conversion.pyx b/pandas/_libs/tslibs/conversion.pyx
index 1104c1f8e67a9..33815aee8b795 100644
--- a/pandas/_libs/tslibs/conversion.pyx
+++ b/pandas/_libs/tslibs/conversion.pyx
@@ -520,43 +520,35 @@ cdef _TSObject _convert_str_to_tsobject(object ts, tzinfo tz, str unit,
         )
         if not string_to_dts_failed:
             reso = get_supported_reso(out_bestunit)
-            try:
-                check_dts_bounds(&dts, reso)
-                if out_local == 1:
-                    return _create_tsobject_tz_using_offset(
-                        dts, out_tzoffset, tz, reso
+            check_dts_bounds(&dts, reso)
+            if out_local == 1:
+                return _create_tsobject_tz_using_offset(
+                    dts, out_tzoffset, tz, reso
+                )
+            else:
+                ival = npy_datetimestruct_to_datetime(reso, &dts)
+                if tz is not None:
+                    # shift for _localize_tso
+                    ival = tz_localize_to_utc_single(
+                        ival, tz, ambiguous="raise", nonexistent=None, creso=reso
                     )
-                else:
-                    ival = npy_datetimestruct_to_datetime(reso, &dts)
-                    if tz is not None:
-                        # shift for _localize_tso
-                        ival = tz_localize_to_utc_single(
-                            ival, tz, ambiguous="raise", nonexistent=None, creso=reso
-                        )
-                    obj = _TSObject()
-                    obj.dts = dts
-                    obj.value = ival
-                    obj.creso = reso
-                    maybe_localize_tso(obj, tz, obj.creso)
-                    return obj
-
-            except OutOfBoundsDatetime:
-                # GH#19382 for just-barely-OutOfBounds falling back to dateutil
-                # parser will return incorrect result because it will ignore
-                # nanoseconds
-                raise
-
-            except ValueError:
-                # Fall through to parse_datetime_string
-                pass
+                obj = _TSObject()
+                obj.dts = dts
+                obj.value = ival
+                obj.creso = reso
+                maybe_localize_tso(obj, tz, obj.creso)
+                return obj
 
         try:
-            # TODO: use the one that returns reso
             dt = parse_datetime_string(
                 ts, dayfirst=dayfirst, yearfirst=yearfirst
             )
-        except (ValueError, OverflowError) as err:
-            raise ValueError("could not convert string to Timestamp") from err
+        except ValueError as err:
+            if "out of range for month" in str(err):
+                # dateutil raised when constructing a datetime object,
+                #  let's give a nicer exception message
+                raise ValueError("could not convert string to Timestamp") from err
+            raise
 
     return convert_datetime_to_tsobject(dt, tz)
 
diff --git a/pandas/tests/indexes/datetimes/test_date_range.py b/pandas/tests/indexes/datetimes/test_date_range.py
index 14bfb14d27239..ec9a6842dc91e 100644
--- a/pandas/tests/indexes/datetimes/test_date_range.py
+++ b/pandas/tests/indexes/datetimes/test_date_range.py
@@ -980,7 +980,7 @@ def test_misc(self):
     def test_date_parse_failure(self):
         badly_formed_date = "2007/100/1"
 
-        msg = "could not convert string to Timestamp"
+        msg = "Unknown string format: 2007/100/1"
         with pytest.raises(ValueError, match=msg):
             Timestamp(badly_formed_date)
 
diff --git a/pandas/tests/io/pytables/test_select.py b/pandas/tests/io/pytables/test_select.py
index 76d5cc5672e40..b0c9b85e7ad05 100644
--- a/pandas/tests/io/pytables/test_select.py
+++ b/pandas/tests/io/pytables/test_select.py
@@ -939,7 +939,10 @@ def test_query_compare_column_type(setup_path):
             v = "a"
             for col in ["int", "float", "real_date"]:
                 query = f"{col} {op} v"
-                msg = "could not convert string to "
+                if col == "real_date":
+                    msg = 'Given date string "a" not likely a datetime'
+                else:
+                    msg = "could not convert string to "
                 with pytest.raises(ValueError, match=msg):
                     store.select("test", where=query)
 
diff --git a/pandas/tests/scalar/timestamp/test_constructors.py b/pandas/tests/scalar/timestamp/test_constructors.py
index 4143e52bbb7ed..aa5a55e6ca39d 100644
--- a/pandas/tests/scalar/timestamp/test_constructors.py
+++ b/pandas/tests/scalar/timestamp/test_constructors.py
@@ -429,7 +429,7 @@ def test_constructor_nanosecond(self, result):
     @pytest.mark.parametrize("z", ["Z0", "Z00"])
     def test_constructor_invalid_Z0_isostring(self, z):
         # GH 8910
-        msg = "could not convert string to Timestamp"
+        msg = f"Unknown string format: 2014-11-02 01:00{z}"
         with pytest.raises(ValueError, match=msg):
             Timestamp(f"2014-11-02 01:00{z}")