If I needed to keep the harvested yeast in the fridge for longer than I needed, could I just make a new starter and leave it to ferment out and then put it in the fridge again. Would this help to keep the yeast healthy?
Yes. Pitching a portion of the yeast into a new starter would certainly produce a revitalized pitch of yeast. Whether or not that makes sense probably depends on how long it's going to be before you plan to actually use the yeast.
The yeast will be most healthy toward the end of the starter's fermentation, so I've known homebrewers who keep stored yeast in a refrigerator until it's needed and throw it on a fresh starter prior to brew day if they feel they've stored it for what they consider "too long." That's a fairly subjective amount of time, but I think 3-4 weeks is a safe upper limit). I suspect your best bet is to rely on some combination of that approach and the one you are proposing. Keep your original pitch as long as you feel comfortable (let's say 3 weeks) and pitch some portion of it into a starter if you aren't almost ready to brew but want to hold on to it. On the other hand, if you plan to brew on the 4th week, you might hold out and pitch some portion of it into a fresh starter a few days before your brew day to optimize for pitching the healthiest yeast possible.
Notice that I've specially suggested pitching "some portion" of the reserved yeast into the starter. If you've reserved most of the yeast cake from a 5 gallon fermentation, you probably don't want to throw it all into a starter; too much yeast is just as bad as not enough. Think of how much yeast slurry you get out of a commercial package and pitch a maximum of 2-3 times that amount into your starter.