Check with an accountant or financial planner to make sure that that's the right approach for you. What you're describing may not be the best option.
If you decide to go ahead, then check with a good mortgage broker.
It is possible to have a mortgage secured by another piece of property, though I don't personally know any brokers who offer that. Another possibility is to set up a HELOC on the property with equity, then use the funds from the HELOC to help purchase the second property. Or you could always look for a second property that wouldn't require any down payment. That wouldn't occur through traditional lenders, but you could buy a property "subject to" the existing mortgage, if you found the right property. Note: A subject-to acquisition will violate a lender's due-on-sale clause, but it is frequently done, especially in today's market. Or you could find a second property and either lease-purchase or lease-option it. And the list goes on. But, yes, there are lots of ways.
Just make sure you know what you're doing financially before moving forward.
I am a Dallas Real Estate Agent and loan officer few fyi's
a) Never purchase a home w/o a realtor involved even if that is from a builder or a for sale by owner
b) Have you taken inconsideration all the factors when you own rental property? I also own rental property if you don't know real estate difficult to work with all the problems and etc.
c) Mortgages are driven by the following;
1) Credit scores
2) Debit ratio
3) Locate a season loan officer that can review your short term and long term goals.
No you can't use a single family home you currently own for a down payment considering you don't own the home out right therefore you still have pending mortgage.