diff --git a/src/mcp/client/session_group.py b/src/mcp/client/session_group.py
index 9610212642..57577c6621 100644
--- a/src/mcp/client/session_group.py
+++ b/src/mcp/client/session_group.py
@@ -147,7 +147,7 @@ def __init__(
         self._session_exit_stacks = {}
         self._component_name_hook = component_name_hook
 
-    async def __aenter__(self) -> Self:  # pragma: no cover
+    async def __aenter__(self) -> Self:
         # Enter the exit stack only if we created it ourselves
         if self._owns_exit_stack:
             await self._exit_stack.__aenter__()
@@ -158,7 +158,7 @@ async def __aexit__(
         _exc_type: type[BaseException] | None,
         _exc_val: BaseException | None,
         _exc_tb: TracebackType | None,
-    ) -> bool | None:  # pragma: no cover
+    ) -> bool | None:
         """Closes session exit stacks and main exit stack upon completion."""
 
         # Only close the main exit stack if we created it
@@ -323,7 +323,7 @@ async def _establish_session(
             await self._exit_stack.enter_async_context(session_stack)
 
             return result.server_info, session
-        except Exception:  # pragma: no cover
+        except Exception:
             # If anything during this setup fails, ensure the session-specific
             # stack is closed.
             await session_stack.aclose()
diff --git a/src/mcp/client/streamable_http.py b/src/mcp/client/streamable_http.py
index aa3e50e07e..73d5efbcc6 100644
--- a/src/mcp/client/streamable_http.py
+++ b/src/mcp/client/streamable_http.py
@@ -467,17 +467,26 @@ async def _handle_message(session_message: SessionMessage) -> None:
                         read_stream_writer=read_stream_writer,
                     )
 
-                    async def handle_request_async():
+                    async def send_message() -> None:
                         if is_resumption:
                             await self._handle_resumption_request(ctx)
                         else:
                             await self._handle_post_request(ctx)
 
+                    async def handle_request_async(request: JSONRPCRequest) -> None:
+                        try:
+                            await send_message()
+                        except httpx.TransportError as exc:
+                            logger.debug("Error handling request", exc_info=True)
+                            error_data = ErrorData(code=INTERNAL_ERROR, message=f"Transport error: {exc}")
+                            error_msg = SessionMessage(JSONRPCError(jsonrpc="2.0", id=request.id, error=error_data))
+                            await ctx.read_stream_writer.send(error_msg)
+
                     # If this is a request, start a new task to handle it
                     if isinstance(message, JSONRPCRequest):
-                        tg.start_soon(handle_request_async)
+                        tg.start_soon(handle_request_async, message)
                     else:
-                        await handle_request_async()
+                        await send_message()
 
                 async for session_message in write_stream_reader:
                     sender_ctx = write_stream_reader.last_context
diff --git a/tests/client/test_session_group.py b/tests/client/test_session_group.py
index 6a58b39f39..165a9c7337 100644
--- a/tests/client/test_session_group.py
+++ b/tests/client/test_session_group.py
@@ -51,6 +51,27 @@ def test_client_session_group_component_properties():
     assert mcp_session_group.tools == {"my_tool": mock_tool}
 
 
+@pytest.mark.anyio
+async def test_client_session_group_context_manager_closes_session_stacks_with_external_stack():
+    class SessionStack(contextlib.AsyncExitStack):
+        def __init__(self) -> None:
+            super().__init__()
+            self.closed = False
+
+        async def aclose(self) -> None:
+            self.closed = True
+            await super().aclose()
+
+    session_stack = SessionStack()
+    group = ClientSessionGroup(exit_stack=contextlib.AsyncExitStack())
+    group._session_exit_stacks[mock.Mock(spec=mcp.ClientSession)] = session_stack
+
+    async with group as entered:
+        assert entered is group
+
+    assert session_stack.closed
+
+
 @pytest.mark.anyio
 async def test_client_session_group_call_tool():
     # --- Mock Dependencies ---
@@ -278,6 +299,25 @@ async def test_client_session_group_disconnect_non_existent_server():
         await group.disconnect_from_server(session)
 
 
+@pytest.mark.anyio
+async def test_client_session_group_streamable_http_connection_error_surfaces() -> None:
+    async def fail_request(request: httpx.Request) -> httpx.Response:
+        raise httpx.ConnectError("offline", request=request)
+
+    http_client = httpx.AsyncClient(transport=httpx.MockTransport(fail_request))
+
+    with mock.patch("mcp.client.session_group.create_mcp_http_client", return_value=http_client):
+        async with ClientSessionGroup() as group:
+            with pytest.raises(MCPError) as excinfo:  # pragma: no branch
+                await group.connect_to_server(
+                    StreamableHttpParameters(url="http://example.test/mcp"),
+                    ClientSessionParameters(read_timeout_seconds=2),
+                )
+
+    assert excinfo.value.error.code == types.INTERNAL_ERROR
+    assert excinfo.value.error.message == "Transport error: offline"
+
+
 # TODO(Marcelo): This is horrible. We should drop this test.
 @pytest.mark.anyio
 @pytest.mark.parametrize(