I have a couple of batches of hard cider going. This is my first time making hard cider. I tested the gravity yesterday after about 3 weeks of fermenting and it's hit "dryness" at 1.01. There is a lot of headspace in the carboy and I want to let it sit for a few more weeks to clear. My question is, can I add fresh squeezed apple juice the carboy to take up the space? I've read water can be used, but it will dilute the cider.
'[C]an I add fresh squeezed apple juice the carboy to take up the space?'
Sure, but introducing fermentable sugars (from the fresh juice) would re-start fermentation and defeat your stated goal of clarification.
'[W]ater can be used, but it will dilute the cider.'
Correct, and unless this is what you want, there's no reason to add water, especially just to fill the head-space.
If you had a healthy fermentation, though, there should be no reason to fill up the head-space at all since the fermentation will have produced abundant CO2 (enough to fill the head-space) and this is fine to age cider under long enough to clarify.
Basically, just leaving it be is probably the best thing you could do. I'm not sure either strategy you mention would be of any real benefit.
If you are afraid you destroyed CO2 layer already, or you want to move cider to new carboy to prevent it sitting on layer of yeast, and you feel you need to fill up head space to avoid oxygenation, buy glass balls. They are easy to sterilize, and can take up volume without restarting fermentation or diluting cider.
I'm not arguing if it's needed, I'm just giving you a way to do it safely.
Note: You can buy glass balls from wine making stores, but they sometimes are also available from general industry vendors, too. Prices seem to be similar, but industry vendors provide better datasheets on their products, like kind of glass, safe temperature range etc. Usually it will give you no real benefit, but if for the same price you may know, or know not, it's better to know in my opinion. Even if your cider is unlikely to reach borders of safe temperature range, ever.
Adding juice will undo any settling and clarification efforts you may have performed.
How much below the shoulder is your cider? If it just there at the shoulder, you are talking about only an ounce or two of water per finished and corked bottle.
What I do is I keep some wine (or cider) in reserve for topping up. Or if it's my first batch of "something", I make 7.5 gallons to start, and when I have rack, I rack into a 6 gallon and an additional 1 gallon jug. This way I have the same stuff to top off with.
It's nice to have a variety of carboy sizes for this reason. Mine list at (in gallons) 7.5, 6, 5.5, 5, 3 and several 1 gallon jugs for trial and error and for that topping off portion I need to keep around.