I'm not sure the 10,000 lightyear thickness of the galaxy is entirely applicable here. Yes, the gas
is about 10,000 lightyears thick, but the actual stars
are only around 1000 lightyears thick. I think. I could be wrong.
So depending on how big you think the Federation is and exactly where the Galactic barrier is, the UFP could very well extend all the way from top to bottom.
If not the UFP then certainly explored space. Rememer, in Enterprise the UFP was just being established.