I've searched all over the IRS website for documentation concerning what I guess would e called a "mixed use" type of secured loan, and I can't find jack squat that comes anywhere close to your situation. But from what I can find in IRS Publication 536 at https://www.irs.gov/pub/irs-pdf/p936.pdf on page 3, it appears things aren't what you're expecting.
From my interpretation, whatever you claim as mortgage interest must be secured by the real estate the mortgage is for. Furthermore, there's a limit on mortgage interest for a refinanced loan. In the case of a cash out, you can only claim the percentage of interest paid on the new loan, that is equal to the percentage of the new loan that was used to pay off the original amount owned on the original loan.
So if your mortgage balance before the refi was $50K and you refi'd for $100K, that means $50K of the new loan (50% of the new loan) was used to pay off the old mortgage. So only 50% of the interest paid on the new loan loan is deductible on SCH A for the life of the loan.
Since the remaining $50K was not used to improve the home that secured the new loan, it's not deductible at all.
The clarity you're looking for, I can't seem to find anywhere. With my current line of thinking, since that $50K you cashed out and used to pay off the rental is not secured by the rental (it's secured by the primary home you refinanced) I believe it's not deductible on SCH E, or anywhere else for that matter.
But I also interpret it that, if you had used that money to go out and "buy" a rental property, then it would be deductible on the SCH E.
Overall, I highly recommend you spend a few bucks of that cash out for professional help. Would also be nice if they could provide you a citation on the IRS website (which you could share here) that clarifies things one way or the other.
