I can think of a couple ways to get the most military use out of the Starship. One would be to design relatively cheap, basic, spy-sats, to launch in large numbers, getting rid of the coverage gaps that currently exist. To be real clever, they could also have Starlink hardware attached, and be secretly launched alongside regular Starlink satellites. Starlink itself has some interesting military potential, and is getting some Airforce dollars already.
Another use would be creating basic GPS satellites that can be tossed up by the dozen if the regular ones start getting shot down.
And now for the exciting part - Starship lifting ~100 tons to LEO is enough to make the "Rods from God" concept viable. The Air Force estimated that a 9 ton tungsten rod would hit with a force of 11.5 tons; figure 9 rods per flight to leave plenty of mass for extra fuel and guidance equipment, and the Starship can dump more than a 100 tons TNT equivalent on a target. In comparison the venerable B-52 has a payload of 32 tons. This would be perfect for destroying any high-value, stationary targets, like shipyards, powerplants, nuclear enrichment facilities, and so on.