If you are an employer...

[user1483390742]

If you had a great part-timer, but needed a full-timer, would you advertise for one, or look for another person to do a job share?

First I'd ask the part timer if they were interested in increasing their hours, then, if they weren't, I'd advertise for another part timer.

Offer the part timer more hours, and if they declined depending on the job I'd either look for another person to work part time (eg if it was a receptionist or something similar) or make the position redundant and advertise for a full time worker.

Yes, would ask the part timer first if they wanted to increase their hours. If not, then yes, I'd probably look at a job share but obviously that would depend on the FTE of the existing part timer and the feasibility of recruiting someone for the other half of the job. It would also depend on the nature of the job itself - some jobs don't really lend themselves to a job share arrangement.