How about this for an idea:
(1) install your newly intended 2.5" SATA3 SSD into the primary 3.5" bay, with a 2.5-3.5 adapter bracket. I believe there's a standard Lenovo accessory part in their "tower storage kit" which allows you to do just exactly this.
(2) move the current 3.5" SATA3 HDD spinner primary drive outside of the M91p tower case, to instead be housed in an external enclosure. I recently had to replace a large homebuilt desktop tower machine (with four internal HDD spinners) into a Lenovo M910t (which only has one 3.5" bay), and used this exact approach to relocate two of the four drives in this way. I opted for connecting the now external drives using eSATA rather than USB, for performance reasons.
(3) To house my now external eSATA-connected drives I purchased two of this very fine OWC Mercury Elite Pro enclosure, which supports both eSATA as well as USB 3.0 connection.
(4) To connect both of these eSATA external enclosures and the SATA3 drives inside them to the motherboard of the M910t, I purchased this StarTech 2-port eSATA adapter bracket. Externally you plug an eSATA cable in from the eSATA enclosure/drive. Inside the case there are two SATA data cable/connectors that you then plug into two available SATA/eSATA controller sockets on the motherboard. The SATA and eSATA sockets are essentially the same, except that eSATA also supports hot-swap of external drives) for this purpose controller sockets on the motherboard. But otherwise you can use any two open sockets for connection to the two cables running from the StarTech adapter bracket. This also comes in a 1-port version but I went with the 2-port method, which I needed.
Seems like that should work. You now have your current 3.5" SATA3 HDD spinner housed outside the M91p case in an external eSATA enclosure. And the enclosure is connected via eSATA cable through the StarTech adapter bracket to an available SATA controller socket on the motherboard.
In its place, inside the primary 3.5" bay, you now have a 2.5" SSD installed with a 2.5-3.5 adapter bracket, and also connected to power as well as a SATA controller socket on the motherboard using the very same power and data cables which prevously were used for the 3.5" HDD spinner you've now moved outside the case.