I've never been a huge fan of direct Lovecraft tributes/followups because I find that they rarely capture the essence of what actually makes Lovecraft good. August Derleth is probably the most famous example of someone writing Mythos stuff after Lovecraft's death, but his work is pretty controversial. He writes from a Christian point of view that re-contextualizes everything Lovecraft established, and he's just not a great writer from what little of his stuff I've read. Of course, Lovecraft is also not a great writer, but someone who isn't incredibly skilled building on something written by someone who also wasn't incredibly skilled can get kind of messy.
I mentioned this in another thread a while ago, but the best contemporary Lovecraftian story I've ever read is probably Gen Urobuchi's visual novel Saya No Uta. I don't know if I can actually recommend it to anyone since there's some reprehensible rape stuff in it. but the ideas are scary in a remarkable way and by the end it's clear that the world it's set in is authentically Lovecraftian. Read at your own risk, but it might be the kind of thing you're looking for.