will changing to solid-state make the bass in my wilson sophias better at low volume?

It should look at PASS 350.8 great bass weight and definition.

Firstly, I have no experience with either your amp or speakers. That said, I notice you are trying for more bass with a  triode amp. I suspect the triode will not give you that bass because they usually cut off the bass in favor of midrange emphasis . So, IMO FWIW YMMV you may be trying to get something a tubed triode amp does not readily give.
Second, I've heard all my audiophile life that SS gives better bass. So you can imagine my surprise when I bought my 1st tube amp  about 12 yrs ago and finding out it did BETTER than the SS I had been using the previous 40-50 yrs. Yes, the bass is more "flabby". But having played drums in local bands for 50 yrs, I can tell you that the tubed bass sound more realistic to what I heard while playing with a live bass player (Classic-hard rock). My latest amp, Sachs Kootenay 120 does great on bass...and midrange 7 treble too. I doubt I could go back to SS now.

“I guess I just need to audition a high-powered SS amp and a (sealed) sub in my system”

@musicslug, Before you buy a sub, try a high quality SS Integrated. I suggest Gryphon Diablo 300 that would pair well with your Wilson’s.

I am also of opinion that no full range speaker is capable of producing deep textured low and mid bass that you get from a high quality subwoofer (REL’s) unless you’re using a full range speakers with built in powered sub (like Vandersteen’s).

You are battling against a speaker efficiency of 87 dB vs 92 dB on larger Wilsons, so you need a higher output tube amp to tame the Sophias.

I use a 70 wpc tube amp on the more efficient models with no loss of bass control compared to a 300 wpc solid state amp I used to run them with.  If you get the chance, try both a more powerful tube and and an SS amp and see what you think.

Hello,
The subs are a no brainer. I have two RELs and they really help out the bass at all volumes. You have to turn up the amp so much to get those divers moving on your front speakers. Not the case with subs. Next is the type of amp. Mark Levs have to be turned up so much before the bass engagement. On the other hand the Ayre AX5 integrated amp provide great bloom at all levels. Last is the preamp. I demoed the Rouge RP7 balanced preamp and it was so great at providing that bloom at very low volumes. This is with a SS amp. There is something awesome about a tube preamp with a SS amp. Like yin and yang. I am auditioning the Rouge RP5 preamp which has the phono preamp. Something very cool. You can turn the Phono preamp input into a line input in case you want to use your own phono stage and it has extra preouts for subs. Rouge is really good. If there is a Rouge dealer near you try out some of their gear. If you are in the Chicagoland area check out Holm Audio. https://holmaudio.com/