However, ideally you should cut the original one out right next to the cone and then glue it back into place when done. You can leave a very small spot attached so it's easier to get back into correct position.
I try to reuse the original dustcaps if they are in good shape but it depends on if it's a woofer that needs shimming more so than others. IF you want to remove and reuse the original dustcap, try using MEK (methyl ethyl keytone) with a syringe around the base of the dustcap. This will loosen the glue and make it easy to remove. I also use MEK to remove old glue from the surrounds on both paper and poly woofers with great success. I learned this years ago from my friend that rebuilds speakers for a living. Works like a champ!
Would a place like ACE hardware store have that? I have some 3M adhesive remover, would that work better/the same?
Don't put the dust cap back! Removing it will in almost every case improve the sound of the speaker drastically. You do not need to worry about dust getting inside the speaker, it is extremely unlikely, unless you pour sand in there. If there are small holes in the cone under the dust cap, you can glue thick but small pieces of paper on the underside of the cone to cover them up. Works like a charm.

I've only re-foamed 1 set of speakers without shimming. A pair of bose 901. The speakers have a relatively short travel, so I felt safe not shimming them. I just glued the foam and pushed the speaker in and adjusted as necessary to avoid any coil rub. I would think when you get into some of the woofers that have a longer throw, you wouldn't have much of a choice other than shimming.
That being said, I believe the most popular method is to use a test tone.
...However, ideally you should cut the original one out right next to the cone and then glue it back into place when done. You can leave a very small spot attached so it's easier to get back into correct position...
Good luck.
Doug
This is what I have done and if your not sloppy with glue you'll never know it was cut. Many people will shy away from buying a speaker with non-original dust caps.
For those who don't use the shims, how
do you ensure alignment?