Second playthrough
 

On my first playthrough I didn't collect all artifacts and unpgrades and I'm now on my second. As long as I don't fire at the final boss in the centre at the end, initiating the battle, can I go anywhere on the map as much as I please? I noticed I'll have to do some backtracking to get some collectibles.

Also, I'm up for getting the two lantern run cheevos online. Add me if you need a fourth.

You don't even have to start a new game after defeating the final boss, just select resume game. Anyway, since you already started a new one, you're right. Just don't fire at the eye and it will leave you alone, giving you the opportunity to get any collectibles you might have missed earlier ( or couldn't get ).