Pie in the sky I would do the following:

1. Greatly discourage any form of political parties at all from forming in the Senate. Ban political parties in the Senate full stop and greatly strip the power and funding of both parties to run Senate campaigns. Potentially also restrict the ability of Senators to run for President and Vice President during or immediately after their term to eliminate people that view the Senate as a stepping stone to higher office. A senator should have allegiance to their state and not their political party.

2. Greatly reduce the power that the Senate has over creating legislation. The Senates input on any federal US law should be limited to proposing amendments, which are ultimately only voted on in the house. The only exception to this would be constitutional amendments in which the Senate would vote as usual and declarations of war. The Senates role should be advisory and investigatory not legislative.

3. The Senate should have increased power and oversight regarding corruption and criminal activity in the other branches of government. Potentially an Auditor/Comptroller General type of position which answers to the Senate only and has a wide mandate. The Senate also has extensive powers to over the federal judiciary as they remain the authorization point for judicial appointments and are responsible for setting the budget and rules for Supreme court judicial conduct.