Popped my left SI joint a few weeks ago......... getting off the couch. How embarrassment.
IF it's a real SI joint injury then you gotta do 2 things.
1/ Get the joint back in place.
2/ Release and relax the muscles surrounding the joint. They are all OMG and WTF and need some tender care, like Justin Beiber in the cage with Bubba the Sheep Lover.
A Physical Therapist will generally not have a clue. Sorry.
A hospital will have even less of a clue. Google South Park and the Hardy Boys.
Go to an Osteopath to get the muscles released and the joint back in place. Again, having a plaque on the wall means nothing. Ask around for long time gym users and find an osteo that either does gym or who knows what gym users need. We don't sit on the couch all day beating off to Justin Timberlake movies, at least not all the time. We go to gym and get all, like, shredded 'n ripped 'n horse strong so we need different treatment. Your average dipshit will never put their SI joint out. They will usually have a lower back issue due to a huge gut pulling on the vertebrae.
After the Osteo has it sorted (might take a couple of visits) go to a sports massage therapist who does Active Release Therapy (ART) massage. Again, ask around and find one that knows what they are doing. They will make you scream for your mother and cry like a baby but will ensure that the muscles are fully relaxed, allowing the joint to settle and get rid of any inflammation.
I did the above and I was all good to deadlift in a week, but gave it two weeks just to be sure.
Oh, and do lots of walks. It helps to settle the joint into it's proper position.