Hi Raymond, welcome to ETO!
+/- 0 to 60V is a very big stretch, especially if it needs to supply 3A all the way down to it's lowest setting.
You mention 0-60 in terms of output, but then mention a 60v input! So what do you really have? You need a few volts overhead so if you want 60V out you probably need 65vdc in, then at a low voltage like 5v you are looking at 3A * 60V = 180W dissipation in the regulator. Which isn't impossible, just very very impractical. And wasteful.
(edit) If you measured the transformer's output with no load you are onto a red herring with the voltage. Load it with something that will draw an amp or two and test the voltage like that.
So I suggest you look at using a buck converter to drop the voltage down to a sensible level, it's a good idea to make it track the output so the voltage across the linear regulator is always the same. There are a couple of designs you can use for this.
I've seen another design on the 'net which used a chopper circuit to reduce the voltage by pwm, which you could also consider.
If you're lucky enough that your transformer has taps for other voltages you could come up with a switching arrangement to change between these.
That must have been quite a beefy amp to need such a big transformer!