Don't you need at least a little thrust ~40 mins later to circularize the orbit?

Obviously solid rockets can't be relit.

Could that be the use of the 4th stage?

Rockets are usually two-staged + satellites. The booster take it to outside of the atmosphere, then the upper stage puts it into a ballistic trajectory, and the payload does the circularization burn 45 minutes or so after the liftoff at the peak of the parabola.

Above is the basic semantics, and it can be further optimized, such as by extending battery power for the second stage to use it for circularization, inserting a single purpose satellite-like pusher device above the responsibility boundary at the top of second stage and payload satellite and calling it the third stage, or just adding actual third stage above second stage, etc.

Adding more and more stages improve performance per Konstantin Tsiolkovsky's rocket equation, but it'll add risk factors and also obviously add more dead weights in electronics and engines and support equipment, so 2-3 stages is usually the good balance between performance and risk/costs. You can have as many stages as you want if you think you can handle it.

Good point! The final stage being liquid would also give them more control to fix any errors with the orbit.