When I attacked the problem of finding a quantum factoring algorithm, I had read four or five papers on the subject, which constituted nearly all the literature on quantum algorithms at the time. However, there were lots of other relevant papers that I didn't even know about in the field which would later be called quantum information theory, and I didn't feel compelled to do a literature search to find them.

This is one extreme. If you try to go this route, you may very likely miss some important techniques that are commonly used in the new field, so I would actually recommend substantial reading in the new field. I had to do that when I started working on quantum information theory.

It would make a lot of sense if you worked on your problem while you did this substantial reading (even though you're liable to go in the wrong direction), because that will be a good guide for choosing which papers you should read. It also helps if you have a colleague in the new area you can talk to or collaborate with. So, as Deane says in the comments, "It depends."