When you are moving around on separate tickets, the airlines are generally going to deny boarding unless you have the right to enter the country in question. Especially if you are wanting to overnight in country (even if it's airside at the airport)

Also note that residence and your location during the prior 10-14 days is usually what matters, not your passport. So if you've been in Spain long enough, you would be subject to the rules for travel between Spain and Hungary, not the US... Even if Hungary didn't accept your CDC card as proof of vaccination (I have no idea of the current rules for Hungary), you might still be able to travel with a PCR or antigen test.

P.S. FWIW you should not be doing multi-country tours of Europe at the moment. Too many moving pieces with restrictions and requirements...