Home Addition loan without enough equity

[Matt Watkins]

I'm looking to add around a $100,000 expansion to my house. I do not have enough equity in the house currently to qualify for a home equity loan to cover this full amount.  I found a Fannie Mae HomeStyle loan where they would calculate equity based on the value of the house after the addition was completed but after calling multiple lenders I was unable to find anyone who uses post addition value. If this were a new house I would qualify for a mortgage to cover what is currently owed plus the $100,000. 

Does anyone have any suggestions on what could be done? I do not want to try to take out a personal loan due to term length and interest. I have around $40,000 in equity right now in the home. 

Thank you

[Matt Watkins]

@Jaron Walling
Value of house: $240,000
Currently owe: $160,000

A new Mortgage at $260,000 would probably be best payment-wise. That's only $20,000 above the current value.  Based on houses in the area with the addition, I would estimate house value at $310k-325k