Is this unprofessional?

I have a large gap between graduation (middle of May) and my start date with the firm I've accepted a full time position with (late October). I plan on studying for the CPA exam and trying to take as many as I can. However, my parents do not financially support me and I will need money during this time obviously. Is it a bad idea to try to get another accounting internship for this summer even though I already have a job lined up?

I was thinking this would be a good idea since I could work for 10-12 weeks, be able to live off that/save so I won't have to work August-October. Rather than finding some retail or restaurant job getting paid minimum wage and hating it.