In Phase 10, what happens when a skip card is played by the only person who hasn't been skipped yet?

You may be misunderstanding the once per round restriction. From the rules:

When someone is skipped, a “round” is once around the table.

So a “round” is not a hand; which ends when a player goes out. It is simply one turn for each player. So if it is your turn, there must be at least one other player who has not been skipped since your last turn.

It is not possible to have all the other players under pending skips when you take your turn. Someone may have been skipped (and are no longer under a pending skip) and can therefore be skipped again, though. There's always a valid target.

For example: the player to your right is a valid target. Even if they had just lost their turn to a skip, having lost their turn, they're no longer under a pending skip, so you can play a skip and select them to lose their next turn. They still will only lose one turn that round, as well. That is: no one else can play another skip on them until they pass their turn in the play order.

Wiki confirms this reading:

A player cannot be skipped twice in the same round; they must lose turn in the round before being skipped again. (They can be skipped twice in a row but not until they miss their turn.)

As do the rules on the designer's page

If a player uses a “Skip” card against another player, no other player can use another “Skip” card against that player until that player’s turn has been skipped.