We know private Hudson (Bill Paxton) was "getting short" because, according to him, he only had "four more weeks" left in his enlistment. However, he doesn't show up on the ship's roster for the marines. It's a little difficult with the scene fade in, but you can see that everyone else is there, even the passengers and android.

Due to some navigational delay with the distances involved, or maybe some issue before they left, maybe the Sulaco took longer to reach Hadley's Hope than expected. In the intervening time Hudson hit his end-of-service date and and was officially mustered out. The ship's computer automatically removed him from the active duty mission personnel.

Not sure what the protocol is for the Marines, but they probably have contingencies for such things when hypersleep and long distances are involved. They probably noted the error and performed a standard mission-required extension.

The real kick, though, would've been if he wasn't legally required to perform his duties as a Marine anymore. It's possible he may have been able to sit out the last mission safely aboard the ship. Without the little extra firepower and his technical expertise, the entire team would've been lost on the planet and he could've ended up as the sole survivor.