I did my exhaust yesterday and it looks great and seemed to fit great from Waldron's.
Most of the hangers were there and I got everything tightened right up and it seems rigid.
I have a slight vibration when I first move car and put it into gear. Once moving it goes away.
One place I found a close fit was on the pipe coming from the muffler that goes into the frame, before those rubber sheets with the hole for the pipe. Those are in poor shape but nothing in there seems to rub. So at that point before that rubber sheet it could touch the frame if I moved it so I cut a piece of heater hose and jammed it into that spot to stop the potential contact with the frame.
The only other places it appears close is on that same pipe coming from the mufflers on each side as the go towards the middle under the floor pan. I cannot get them to move easily there but is the engine torque enough to move them? I think I will try some more rubber heater hose there. Jam some between the pipe and floor pan.
Is the heater hose back behind the muffler towards the rear strong enough not to melt or burn?
I'm not sure how close the '59 and '60 are but I would hazard they are probably pretty similar. I went with Waldron's too and everything fit very well.
I did the exhaust on my '59 and you can see the details here:
https://forums.cadillaclasalleclub.org/index.php?topic=169595.0
I had to replace all of the missing original mounts. Part of that was the rectangular rubber piece that the pipe goes through. There was plenty of clearance around it. Looking at the photos, there was also clearance around the area where the pipe closely hugs the floor pan. It is not a lot but is enough not to rub anywhere. I'd suggest loosening the pipes and see if you can tweak them a bit to get any extra clearance.
Yours may just need a little tweaking. Or perhaps the old mounts are out of spec and are allowing the pipes to be in positions that they should not be. I don't have any first hand with old mounts as all of mine are new reproduction.