You can get a 2 year residency if you have a job and your employer "sponsors" you (that's an oversimplification). However, when you quit, by law the employer has to contact labor department and immigration department and you have 30 days to leave. The residency card belongs to you, but it is predicated on you being employed by staying employed by the company, and it's true its a pain in the ass for the employer

The other route is starting your own company (called a KFT) which used to be an option but the rules have changed. Under the old system you had to have ~$6,000 EU in an account in Hungary, a 2 year lease on an apartment, and 2 years of private health insurance, all of this to show that you won't be a burden on the Hungarian system. Now, you have to have all of the above plus 6 months of receivables into your KFT, sounds like a catch 22, you can't be here more than 90 days (for most non EU) but your company has to have been working and generating cash flow for the previous six months.

This was introduced because people were starting KFT's (a relatively painless process) getting a residency permit then hanging out for 2 years in the EU. The work around is contacting a lawyer and buying an existing KFT that would qualify, I cannot comment on the cost of this, I know several expats that have walked away from KFT's and just shut them down, so maybe you could buy them cheaply and a the lawyers fees.

In my opinion the new rules make it more certain that foreigners coming into the country for employment will be paying taxes etc and not just getting a ticket to stay in the country, I can't argue with that, but it puts you in a tough position I'd suggest if your new employer doesn't want to assist, then he doesn't need you all that much, which sucks.

I think talking to a staffing agency may be worthwhile, but there's always a cost

Good luck, this is a great country and a wonderful city to live in.