Softmod only supports drives that can do the locking thing. While it is part of the IDE spec not all drives will.
http://xboxdrives.x-pec.com/?p=list has a nice list you can search.
TSOP/chip supports any drive as they don't need to lock. Recall that TSOP models of xbox are the older models and the last ones made will need a chip.
Most SSDs you will find today are going to be SSD in nature. IDE/pata SSDs did exist for a hot moment but they are rare these days, usually very small and probably quite slow as these things go.
SATA to IDE/PATA adapters do exist. I am not aware of any that support/emulate locking, and if there are any then you will have to look for very specific model numbers (no such list or anything). Someone in the future might have a look at this sort of thing similar to how we see optical drive emulation devices being made for all manner of other systems this last few years but that would be the future.
To that end you would have to couple a chip/TSOP mod with an adapter and cram it all in the case (there is some space for such things but some of the adapters I have seen might see you struggle a bit, though in this case with a 2.5 inch SSD that might offset things).
Said adapters are not typically the fastest things on earth either (it is kind of a niche thing) so I don't expect much in the way of speed boost or anything, and frankly loading things from a hard drive is already prone to considerably boosting things over what a DVD (or even worse a file by file dumped ISO) would achieve.
If you have a stock of such SSDs for some reason then yeah, otherwise a decent IDE drive should still be reasonably available, and you can probably pick up a few IDE drives of decent nature, set them up ahead of time and set up a thing to auto copy your saves and whatnot whenever it sees the xbox on the network.