I didn't find a photo but here are some things that could help. I grew up in Pocatello, my second son was born there but they've remodeled many times and it doesn't look like it did back then. Last resort for me would be to contact the librarian and ask for help but I don't know if they do that.
It was called St. Anthony Mercy Hospital. There is an old town email contact that you may be able to ask for a photo:firstname.lastname@example.org
and there is a timeline about the hospitals in Pocatello with and interior photo, nothing exterior: http://www.portmed.org/historic-timeline/
,you may be able to write to them and ask if they have historical photos. Also, I searched "pocatello historic images" and found some links you may want to check them out.
Good luck, Nancy