[Aside to the discussion]
Assuming ANY "spiritual" significance of the jailer's query is questionable. It is because of the way the verse is translated.
Having no Greek skills whatsoever, I would be interested to know what you see in the translation of the verse that questions any 'spiritual' significance to the jailers question. Not disputing you at all, just would like to know what you see (or don't).
....The jailer, who had been asleep and not in the jail when Paul and Silvanus were singing and had NO spiritual interest......
This is an assumption. I'm of the assumption that the jailer DID have a 'spiritual' interest because he had NEVER before seen such great faith and joy that Paul and Silas were exhibiting under such adverse conditions, and was deeply impressed and moved by by it. The earthquake, destruction to the jail, and subsequent fear for his life would have been 'the straw that broke the camel's back' which drove him to Christ, jmho.
(99% of use of this Greek word in literature is talking about delivered from death, danger, etc)
Again, I have no Greek skills, but I totally agree with this. 'Sōzō', translated 'save' (read deliver), is ingrained in most minds to always carry eternal significance and is mistakenly believed to be always synonymous with the birth from above or the acquiring of eternal life. It is not.
I believe the jailer knew enough about this religion of Paul and Silas to have connected that word 'saved' to describe a devotee of the religion of Jesus Christ. IMO, to paraphrase the jailer's question:
“Sirs, what must I do to join this religion (and have this great joy and confidence that you have)?”
I seriously doubt that 'saved from eternity burning in hell' was on his mind.