I agree with the suggestion that several have made about providing an extra litter box. Sometimes they have very strong feelings about sharing, or there may be a signal from the earlier cats that the new one isn't welcome in their bathroom territory. Sometimes it has to do with one or more of them having intestinal parasites, and the vet will be checking for that. You could ask the person you got him from what brand and "flavor" of litter she used, because sometimes they're just picky if it doesn't smell exactly like what they're used to. If you don't like her choice, you can start with it and gradually mix in your preferred one.
Try hydrogen peroxide for cleaning the areas where he's gone (test first in a hidden spot because it might bleach!). Once they have marked a spot with their aroma it's very hard to break them of the idea that it's a good place to go.
The two cats I had until a few months ago didn't mind sharing the litter pan, but I learned after one cat died that they each used only one area of the box. It was weeks before Pogo would go in the back of the box that used to be Bee's area, and she still usually does not go there. Years ago when we introduced a third cat, one of the two earlier cats resented sharing the litter with him and was going in a corner behind the water heater, which was a very hard area for us to reach! We put a new litter pan near there and she straightened out her act.
You could ask the vet about using calming collars or pheromone mists which are supposed to help cats get over anxieties.
Good luck with your new fur baby!