I'm job hunting right now and turning over a lot of similar questions, about how much I should be asking for.

A few years back I got over 80% by switching sectors - I was underpaid at a public sector job I loved, and switched to a private sector job in the finance industry and a higher COL area. Similar to you, they offered more than I asked for because corporate had specific pay brackets for that position.

I think your pay depends a lot on the specific area/tech stack you're working in and who you're working for. Some tech stacks just pay more on average than others, bigger corporations can usually pay more than smaller companies, and private sector will always pay more than public sector (but usually with worse benefits). You can check Glassdoor or similar sites to see what people with a similar title make at the company you're applying to, but that's only helpful at really big companies where there are enough employees reporting to give a good average.

I'm fresh out of grad school, and the shift from 22k to 60k has been life changing

I got a 35% salary increase when I started a new job a few months ago. I was very excited.

Jumping jobs is a great way to increase your salary, but be aware that having a bunch of short stints is a major red flag in the hiring process. You may ace an interview, but who wants to hire somebody and train them with their products and db schemas, just to have them leave in a year.

~220% total comp for me last year, switching companies from a job with okay but below market pay, and becoming a senior software engineer in the move. I think I can feasibly double one more time if I try, but it'd be a bigger push and likely involve working for FAANG. Anything more that is outside my reasonably likely career path.

Last Spring all the Paramedics & EMTs got an hourly increase. I was expecting maybe $1.00-1.50... very shocked with a $6.30hr raise!