Your situation reminds me of mine when i was 1st learning.
My 1st teacher was this oldish dude who played guitar. Not a jazz player by any means, just your typical coverband guitarist, a very solid one though.
For the time that i was with him i probably picked up lots of bad habits, like playing soley with a pick etc, but i loved it, because he actually had me playing songs, and it made me feel asthough i was a musician.
But after 6months or so, i knew it was time to move on. So i found an excellent "bass" teacher, and thats where things suddenly became more challenging and fun. But i doubt i would have progressed so fast in the 1st couple of years had i not learnt to "play" from my original teacher 1st. I reckon if you're a newbie to ANY instrument, having to sit down and learn scales and whatnot before you can even play a song would be dead boring, and interest would wain fairly fast i reckon.
From checking out the songs you're playing in your cover band, you can obviously already play a little dude....They're similar styled songs to the ones i was playing before i got my 1st 'real' lesson. But i would definitely recommend moving on, and finding a specialist bass teacher. But dont just settle for the 1st one you find....do a little research by asking around music shops etc to find who has the best reputation in your area. You want a guy who has a good technique, and who is a compitent sight reader. I say "compitent sight reader" because these are the guys that usually have mountains of transcriptions, charts, and other bass related material for you to work on, that will help you progress. I was lucky in a way, because my teacher was a professional transcriber aswell as bassist, so he had literally thousands of charts, and every "tower of power" bassline ever played trascribed.
So in a nutshell, i reckon you should look for a teacher who:
*Is a strong sight reader, and has fairly strong emphasis on this point....DONT go to a guy that teachers you stuff from tAbz!!!
*Good technique ......you want someone who can iron out your technique, and make it as 'user friendly' as possible.
*Has a strong background in jazz, and jazz theory.
*Make sure he's someone you can get along with ......you dont want some old geek who will cringe if you swear.

(im worse than a sailor, so this always helps)
*Make sure he's got plenty of material .....charts, books, transcriptions etc etc
*Make sure the cat can play ......you want to be impressed by. If you look upto your teacher as a player, it will make you work harder on the material that he gives you.
$0.02