Mt. Saint John & Others
Chip,
First of all I have never climbed Mt. Saint John. However, I have hiked up Hanging Canyon three times in the last few years. I can say that Hanging Canyon is perhaps my favorite spot in the Teton Range; very beautiful, and seldom visited. I have seen Mt. SJ from the shores of Lake of the Crags and it appears to be an acheivable summit with little difficulty. I would, depending on snowpack, consider carrying an ice axe. Inquire at the Jenny Lake ranger station prior to your trip. The rangers at Jenny Lake are extremely helpful, and always enjoy a good chat about hiking and climbing.
One previous poster had an interesting proposition. If you are willing to expand your horizons Buck Mountain is also a wonderful summit, and acheivable as a day trip. I scrambled Buck Mountain via the exciting Souteast Ridge, considered class 4.
I have also climbed Mount Woodring in the Teton Range, which is accessible via Holly Lake and Paintbrush Canyon. Woodring also provides interesting views both of the Grand Teton Massif, and Mt. Moran.
I have access to the climber's guide to the Teton Range, and am willing to share information from that book when you settle on a climb.
I have some photos of my 2006 trip to the Teton Range which includes Hanging Canyon, Buck Mountain, Mt. Moran, and Grand Teton if you wish to view them. They are located here:
http://www.pbase.com/pudgy_groundhog/gtnp06