Are you sure you have a C-clip rear end in that '59? Pretty sure all the '55 -'64 8.2" Chevy 10-bolts with the drop-out pumpkin all had bolt-in axles with pressed-on axle bearings. Once the bearing (or crush collar used to retain it) loses its press the axle walks out. Had a '57 Chevy that was that way some time back, had the original rear in it. Either way, best of luck.
What he said . 65 and newer full size Chevy had C clip Easy way to tell WTH you have for a rear, is that a C-clip rear has a bolted back cover. A 59 would be a drop out center, bolted from the front, and has a welded rear cover.
With a drop-out pumpkin there would be no physical way to use C-clips. You're in need of a new bearing and the assistance of a shop with a press, if the axle itself isn't ruined.
Cool, glad you're back on the road. Hope the rest of your trip goes well without any broken parts. Have fun!!