혹시나 싶어서 = 혹시나 해서 => (just) in case / on the off chance / on a hunch.
혹시 means "by chance", "on the slim chance that (something) might ...", or "should" as the future subjunctive.
Adding 나 to it puts an emphasis, so it's talking about a really small probability.
싶다 here is remotely related to its usage in ~고 싶다, but has a different meaning here. It means having a hunch or an inkling of suspicion about something, not a desire as in ~고 싶다.
- 혹시나 싶어서 뒤를 돌아봤다 - I looked back with a hunch there might be someone there (혹시 누가 뒤에 있나 해서 ...)
- 혹시나 해서 물어봤어 - I asked just in case / I asked spurred by a hunch.
- 혹시(나) 네게 무슨 일이 있나 해서 - In case something's the matter with you. (full form which includes what the suspicion/concern is about)