Can I apply for a multi-entry visa at any immigration office if I have a Non-B visa with a one-year extension?

That's a good question, some immigration offices will not sell re-entry permits to people on extensions from another office, BUT then again, some will.

I'd say you're out nothing by trying to see if you can get the re-entry from the immigration office that serves the area where you are now.

Worst case, if they say no, you go back to the office that issued your extension and get it OR you buy one at the airport before you fly out