Applying to firm twice

I am a freshman and I thought I had a finance internship locked up at a decent sized bank in my state but the SVP is unsure if he can get me in. Now I am thinking of just cold emailing wealth management companies near me and hoping to get something out of that. Would it be bad to cold call /email local PE firms as well, if I intend to try to get an internship with one of them next summer after sophomore year? I don't want to give them my resume and then they never consider me since I don't have much experience to work well in a PE firm. Help me out. thanks