Good morning, John, and thank you for writing. It appears that you are currently at exactly 80 percent loan-to-value. Definitely time to dump that MIP!
I think that your idea of refinancing out of your FHA loan to get a 90 percent conventional (non-government) mortgage is the right move. And there are lenders that fund second mortgages to 90 percent of your property value.
However, you don't necessarily want the same lender doing both, unless you'll get a significant break on the costs that way. It eliminates your ability to compare rates and terms for the second mortgage.
The main thing to consider is that larger loan, compare quotes for that refinance, and contact a couple of the most competitive lenders. That's the time to ask about the second mortgage -- check the rate and see if you actually save on title and appraisal charges by going this route.
I do have a call in to a title company, because I know that if you use the same title company, you may not have to use the same mortgage lender to get a break on your closing costs.
Finally, if you disclose that you're going to pile a 10 percent second mortgage on top of the first mortgage, the lender might consider you more risky when underwriting. And I do know that Fannie Mae does add a surcharge for a combined loan-to-value (CLTV) of .25 to .75 percent. https://www.fanniemae.com/content/pricing/llpa-matrix.pdf