Is the level 3 fleece jacket for garrison or field wear?

DA PAM 670-1 states that the fleece can be worn as long as it is authorized by your commander. No where does it say that the NCO in the parking lot makes policy. So as previously stated by the 1LT Martin and LTC Polichette, you are good to go unless a policy otherwise states so.

Yes, you can wear the fleece in garrison, as outer garment... Well, let me elaborate... generally that is governed by local policy if not covered in 670-1... Why would it have name, US Army, and rank on it?

I've always heard the opposite, in that the fleece was for garrison wear only. For whatever reason a lot of CSMs and 1SGs dislike the fleece and end up making their own arbitrary policies. If the Commander signs off on it there's not a lot you can do. You also may have to pick your battles. Getting into an argument with a SFC over a garment probably isn't going to end well even if you're right.