Of course you can do it, this is Dwarf Fortress!
There are two different ways to separate old clothes from new clothes. Which one you use depends on whether you are trying to sort clothes that are already combined, or if you are trying to keep them separated as they are manufactured/used.
Keeping newly made clothes separate from old worn clothes:
Normally this could be accomplished with two stockpiles, however because of the bin bug if you want to store the worn out clothing in bins you need to use three stockpiles. It is pretty simple.
Step 1: Clothier -> Stockpile A
Stockpile A accepts clothing (obviously), is linked to t)ake from the clothier and is set to a)ccept from links only. Now this stockpile will contain only brand new clothing that comes out of the clothier. No other clothing will be put here. This stockpile can be set to allow bins or not, depending on your preference.
Step 2: Stockpile B
Stockpile B is a clothing stockpile that a)ccepts clothing from anywhere. Because of the bin-bug this stockpile must have bins disabled. Now all the other discarded clothing in the fortress will be carried to this stockpile. Voila, new clothes in stockpile A, worn clothes in stockpile B
Step 3 (Optional): Stockpile C
Stockpile C is optional and only used if you want to get the worn clothing into bins without triggering the bin-bug. Stockpile C is a stockpile that allows bins but is linked to take from Stockpile B and set to a)ccept from links only. This is the stockpile that would be located next to your depot.
The general concept to avoiding the bin-bug, is that any stockpile that allows bins cannot be an "a)ccepts from anywhere" stockpile. It must be restricted to "a)ccepting only from links". Otherwise the dwarves will pick up a bin half-full of XTroll fur sockXs, march over to your new clothes pile, fill the bin and carry it back to the old stockpile. Very annoying.
How to separate new clothing from worn clothing after they are mixed:
There is a little labor in this, but it works. Let's assume you have a bunch of bins that have a mixture of new and worn clothing in them. Here's how you can separate them out again.
Step 1: Get your dwarves to d)ump everything. Build a dump in a convenient spot and disable all your other dumps. Mark all the items for dumping. If they are in bins, you can use d-b-d which will mark the bins and everything inside, and then k) over the bins and press d) to unmark the bin itself. Your dwarves will empty all of the bins/stockpiles to the dump and carry the empty bins away to wherever bins are stored. Note: it is important that you unmark the bins themselves, otherwise the dwarves will just carry the full bin to the dump without emptying it.
Step 2: Getting the new clothes out of the dump and back to Stockpile A. This is a little tricky so be careful. Temporarily break the link between Stockpile B and Stockpile C. (Remember, Stockpile B accepts clothing from anywhere but does not allow bins.) Ideally Stockpile B should be empty when you do this, but it's not necessary. Now link Stockpile A to take from B. Then k)ursor over the dump and go through it and un-forbid all of the new clothing. Each piece will be carried to B and then subsequently to A. Wait until all the new clothes are out of the dump (or until A is full). Unfortunately, any worn clothing that happens to be discarded while this is happening will get caught up in this cycle and also moved to B and then A. So put the links back the way they were, and then go through stockpile A and re-dump any worn clothes. When your dwarves are finished with that, all the new clothing will be back in Stockpile A and all the worn clothing will be forbidden in the dump.
Step 3: Getting the worn clothing out of the dump and into Stockpile C. This last part is easy. Make sure you put the links back the way they were before Step 2, so C takes from B again and A only takes from the clothier. Now reclaim all the worn clothing in the dump and your dwarves will move it to B and then C.