When working with multiple contracts, what is the best way to track expiration dates to prevent the contract from expiring before project completion?

You can use Document management software like openkm which is open source and has facility to set document expiry.

Simple way is to create a master scheduled to track all the time bar (or money) i.e. expiry date, insurance PG validity, etc and put a script for email reminder in advance say3 months before.

Of course the best is to have system generated notification i.e. SAP etc.

To hold regular meetings and assess the time line and job times