Requirements for coggins and health certificates vary from state to state.
Here's a decent summary by state: Transportation Requirements by State
Some states have a lot of additional requirements, ie, if traveling thru Illinois you DO NOT need a health certificate if the horse never touches state soil. BUT, you must have a valid, negative coggins.
Some states require entry inspections. This is true of most states that have brand laws. All equines and mules entering the state are required to stop at the port of entry(usually a weight station).
Some states require a CDL license if you have any markings, signs, etc that refer to your ranch, farm, etc. on the truck or trailer.
IN SUMMARY, before going into a state you are not familiar with, check with the DOT and State Vet. AND keep both phones numbers available to you. Some state inspectors really get carried away.
And DON'T pass a state inspection facility, the results are both time consuming and very expensive. Imagine having your stock quarantined until such time as the state, your in, draws new blood and gets a negative coggins back from the lab. Then spend 30 days after that in quarantine. And your truck and trailer impounded until your court case comes up, and you pay the fines. NOT worth the risk.