Hi there and welcome,
Sorry to hear about all the night wakings. You mentioned he was sleeping longer stretches until 3 months old. Did anything change around that time? Is that when you introduced formula? Oatmeal or rice cereal? If he falls asleep but then wakes soon after that, it sound like he's waking up out of discomfort.
I am also wondering if he is having too much daytime sleep - which can sometimes lead to frequent nightwakings. At 5.5months, most babies can manage 2.5hrs A time and do well on 2x1.5hr naps and a catnap. So you could look at stretching his A time (go slowly and add 10-15 mins every 3-4 days so that he doesn't get overtired) and shorten the am and pm nap to 1.5hrs so that your routine would look something like this:
6.30am - wake
9-10.30 - morning nap
1-2.30 - afternoon nap
5-5.30 - catnap
7.30 - bedtime
What do you think?