Can you transfer from a Non cap H1b to a regular H1b without going to through the quota?

I’m looking for non profit jobs to apply for cap exempt H1b visa since i graduate soon and wont be able to wait until April for the regular H1b process. I wanted to know in case I get a H1b through a cap exempt employer can I later transfer to a regular private company by usual H1b transfer or do i have to go through the regular April H1b and subject to quota restrictions?