The best and softest 2.5" lift for your '62 is the Skyjacker Softride.
I installed this kit on my '71 back in '95 or so, and its still on, working well. The flex is amazing, as good or better than a stock (disconnected) TJ.
The '71 has the same springs as your '62.
(I think Skyjacker calls it a 2" lift now, vs. a 2.5" previously)
I also upgraded to poly bushings, and stronger shackles.
Note that if your rig is heavier than stock the Skyjacker will sag a bit over time. I compensated for this by using rear shackles longer than stock, to maintain a level ride.
I ran 33x12.50 MTs on my '71, and Rancho 9000 adjustable shocks. When offroad I set the shocks to "0"... so as to not restrict articulation. My rig rode softer than stock TJs on the trail, as attested by many of my passengers.
Quite a few of the guys at Early CJ5 forums (specifically for 1955 - 1975 CJ5s) run the Skyjacker Softride. You should check out that forum for all the info you'll need for your '62.
Another brand to consider is BDS. This is a good product, flexes well, but is not as soft as the Skyjacker.
Oh BTW, when you remove the old rubber bushings from the spring and shackle hangers, you will find an thin outer metal sleeve rusted inside the hanger. This sleeve was around the rubber, and must be removed before you can install new bushings.
They don't want to come out.
Hammer and chisel can take hours. Air chisel works well. Torch is quick if you have one and you are very careful/precise.
The best method with hand tools is a hacksaw.
Take the blade out of the saw frame, and put it through the hole/sleeve.
Then put the blade back in the frame with the blade still in the hole.
Make two cuts in the sleeve, about 1/4" apart. Then the sleeve will pry out easily.
This method takes about 5 minutes per sleeve.
Nice looking '62 !
Good luck with it.
Maybe see you over at ECJ5...
Oh, here's a pic of the '71, when I owned it (sold it to a friend in '07)