2200-2300, we're talking mostly generation ship technology. Yes, there were primitive hyperdrive-powered ships by then, but they were small and dangerously unreliable - explorers might use them, and the fledgeling Federation would have operated a hyperdrive-capable Navy to enforce the Federal rule on the colonies, but if you're founding a colony (or moving a colony), and you want to be assured of arriving intact, you're pretty much restricted to travelling at generation-ship speed of 0.2c.
There's not much space within range at the start of the century. The first generation ship reached Delta Pavonis in 2180, for example. At the other end of the century, the colonists of Achenar arrived by primitive hyperdrive-capable colony ships by 2310, arriving at what was then considered an extremely remote star system. So if there's a mystery colony founded somewhere during the 2200s, it will have been founded somewhere in the shell in between the distances of those two star systems. Thousands of generation ships are said to have been launched during this time period.
If I had to guess a direction, I would guess in the general direction of Lave. They're called the "Old Worlds" because they were founded much earlier than other colones at similar range from Earth. Something must have been pushing colonization in that direction.