Nope. Out of the caste.
Out of the caste or straight to solahma status? I think it's the latter but I have to admit I don't feel 100% sure on this one.
My evidence:
-The Diamond Sharks letting warriors leave the warrior caste is perceived as this unique thing (some of those were successful, bloodnamed, others were not, and letting them do so was like an extension of hegira). That suggests that's not how it works in other Clans.
-Within Clan Cloud Cobra, we get EcKhan Jali N'Buta, who effectively retired from active duty to run the Tongo Cloister, yet is still a part of the Clan Council and has influence over politics. That implies that Jali is still part of the warrior caste.
-Cyrilla Ward, respected Bloodname House leader. They won't even give her a 'mech to go die in, she chooses suicide. Yet she is clearly still considered part of the warrior caste. She is respected, she has just lost that "active" warrior status.
-In TRO 3085 in the Clan Ghost Bear infantry section we are shown the Clan assault conventional infantry (a variation of solahma infantry). The fluff on them notes that because the Ghost Bears field such a large elemental force, that there is a continual supply of "retired or failed" elementals. They are then outfitted as these infantry, which seem to work with the Watch and are regarded as part of the Clan's internal security force.
-Also in TRO 3085 in the Hell's Horses section it notes a specific infantry unit that are typical of the Clan's ad hoc infantry points, made up of test-downs that have become solahma.
So in a few pieces of evidence we see the terms "retired" or "test-down" then translated into "solahma".This suggests to me that if a warrior loses their warrior status for any reason, it's straight to solahma. Whether it's at age 25 or 40. Whether they are bloodnamed or not.As for bloodnamed or not. I could see the bloodnamed getting treated much better. Like that category of solahma that seems to retain mechs/elemental suits/fighters and are assigned to solahma clusters and then committed to battle to find that honorable death. Assuming they haven't landed some other post, like leading a Bloodname House or a Cloud Cobra cloister.
It is worth noting, you don't see a lot of solahma Khans and solahma Galaxy Commanders. It seems like the Clan is a little more willing to expend energy to let these warriors find an honorable death. It wouldn't surprise me if such warriors are not only held up longer and allowed to retain equipment, they are also put into situations where they'll find that honorable death. It probably helps that such individuals have more political power and allies to help them.
If you don't have a bloodname and end up retired or test down and landing in solahma status, it's sorta like you've lost respect. You aren't respected anymore, your wishes and preferences don't matter. You go where the Clan sends you and you have little recourse. They may not even have to respect a Trial of Refusal or Grevience over the choice of assignment, because they don't have to acknowledge solahma. Conventional marine or solahma infantry/police, wherever you get sent you don't have a choice.
But such individuals are still part of the warrior caste. They just aren't respected the same way as others, and that's where "solahma" becomes a derogatory term.
I do think that, as has been hinted at in this discussion. That there are tiers of solahma. There is clearly a difference between that retired warrior who is made a military governor or a Bloodname House Leader, and that warrior who retains a 'Mech in a cluster (by whatever name, solahma, PCG etc.), and that other solahma who loses access to a cockpit and gets handed a rifle and infantry equipment.
It probably has to do with what level of status they achieved prior to that point, as well as how they ended up solahma, as well as the state of their physical health and abilities. I can also see it having a lot to do as well with the trothkin they have made, the allies they have cultivated, who may yet have some sway in getting them into one of the better assignment options.