I was going to say, absolutely, should be no issues. I have totes full of mag plates in my basement though, and was bored, so went down to investigate.
Almost all of the mag plates for your '59 will have 580210 stamped on them. Now, some have a dash after the 580210, such as a D for example in one case. Some of those, will be missing mounting holes for the cam for example, or other mounting holes for plug wire clips, etc... Almost all were exactly identical though.
Same for the 580360 plates from '61. Almost all looked identical to the '59 plates. I did find a couple though, with same numbers stamped on them, that were different. Different style cams, with different mounting holes for example.
There are some as well, with deeper bulbous holes or cavities, where the coil sits, than others. If you try to use one of these, when you tighten the flywheel, the cavity portion of the mag plate will catch on the powerhead, locking it up. That one had me scratching my head once. Same part number cast into the mag plate.
So, I likely didn't help you understand any better what will work for you, I apologize. I can say in almost all instances you will be fine though. Just to note, there are some rare examples where it will not work.