Purchasing home in cash, then get mortgage?

[Spencer Crist]

I'm looking at my first rental property. Asking price is down to $40k, and the house was sold in 2012 for $100k. The house was rehabbed in 2016. There is a long term tenant in place paying $800 per month. Cash flow will be 200-250 depending on how I structure the financing. 

Instead of investing 8-10k in down payment/closing costs, my thought is to offer $35k cash, and then get a mortgage out to recoup my dollars and move my cash to the next investment. 

I've read that DFEs base the loan off of the purchase price, so that wouldnt help. Cash out refi has a seasoning period which is fine, but I get stuck on the term refinance considering I dont have a mortgage to refi.

Line me out biggerpockets! I appreciate the communities input.