The goal here is a) to demonstrate a history of income, and b) to demonstrate that you have assets (financial or physical), which may back your investment.

@Bryan Richard  You might be able to refinance into an alternative loan product.  

What are some good tenant screening apps that will provide complete reports? the one i am using seems to return less than helpful reports. (not going to name it)thanks!

Then, when it’s time to purchase the third property, you could refi out of the first FHA loan (if you have enough equity), and then use your FHA loan again, or use a conventional loan product. 

You want to reno for the market.....so you want the best product at the best price.   

@Eli Sorey - I think that this answer will vary depending on the production of the agent. 

Cash flow is stable if you choose the right product and manage well.

Mortgages take lien positions and the product you're asking for doesn't exist.Contact a portfolio loan broker. 

Then you can develop investor personas and really hone and refine your marketing campaigns to attract the best investor candidates for your product type, duration, etc. 

I joined the brokerage that was the #1 in my market (by far), then I joined the office location of that company that had the highest production per agent, then I copied everything the top agents in that office did.