A new check on the Congress?

It just struck me that the president doesn't get to exert his check on the Congress (the veto) unless the bill in question is passed by both houses. Considering that the Senate is much more reactionary than the House and that it has its own tools to obstruct bills from ever getting to the president's desk, should we start thinking about providing the president with another check on the more reactionary, less representative Senate when they kill bills?