I would just lay on the compost and perhaps give the areas a watering after spreading the compost.
There shouldn't be much of that mulching left by now, if there is then I would brew up a compost tea with some molasses added and use that to
water once I spread the compost.
Mulch that is thin should be mostly incorporated in around 6 months or less if there is good soil biology already in place.
If you still have most of that previous mulch hanging around, that indicates low numbers of soil microbes are present.
Redhawk