Call the person at the second job and tell them about your situation as you've basically described it here. Let them know that you're definitely interested in interviewing with them about the position they're hiring for, and ask if they can schedule an interview for you on either another day when you're off work from the new job, or perhaps an hour or so after you've gotten off work at the new position.

If they're totally unyielding on working with you on this, that might be an indicator that they're not the kind of company you'd like to be working for, anyway.