A couple weeks ago I brought some geraniums back up from a cool, dark basement to get them started growing again. A couple of them had a few tender green stems several inches long. When I cut them back to pot up I just threw the leftover small green pieces of stem into some water. They began growing almost immediately and now they are rooting as well. I am about to pot up a couple of them. I still have a couple larger, thicker 2+ year-old geraniums dormant in the basement that are still full of tender green stems. I could probably start a bunch of new plants with those.

You can grow plenty of cuttings from your old geranium. You can grow as many cuttings as there are growing tips and it won't hurt your mother geranium plant one bit.

While I grow a few geraniums from seed every year most of my plants come from cuttings taken from plants I overwinter. I have been doing it for many years and have tried all different ways and this is what works for me.

First you have to know that all branches on plants and all roots on cuttings come from a leaf node(joint)

TAKING GERANIUM CUTTINGS
First you must make sure everything you use is very clean.
Take your cuttings: With a sharp knife or razor blade cut a 4 or 5 inch piece from the growing tip, cutting just below a leaf. All roots and branches come from a leaf joint. Remove all but the top 3 or 4 leaves. Let your cuttings callus over--Lay them out somewhere on newspaper. Mine sit on my spare room floor. Let them sit for about 5 days. The leaves will look wilted and dead but they're not. The cut end will shrivel and dry. Letting them callus over and watching the water cuts down on blackleg a deadly fungal disease especially to cuttings.
Now plant your cuttings. In a clean pail mix your soil with the water so you make sure the soil is damp right through. Use a soilless mix like pro mix or one that is formulated for seeds and cuttings. Just make sure it has no fertilizer in it. You want the soil wet but not too wet. Squeeze a handful and some moisture should come out but not a lot. Fill your pots and make a little hole in the center with your finger or something else that will make a small hole.
Dip about 1/2 inch of the cut end in rooting hormone. It comes in powder or gel, either is OK as long as it's for soft wood cuttings. I use Stim-root NO. 1. After it's dipped put it in the hole and cover it in, and you're done.
Keep the soil moist but not overly wet. They will look wilted at first but that will change as it starts to grow roots. You will know it's rooted when it starts to show new growth. If you are going to plant it outside, wait until it is rooted well before you set it out.

Do not skip the callusing over. It is one of the best things I have found to keep blackleg away and do not overwater.

Once you have taken your cuttings trim up the mother plant so it has a nice shape. you want cuts above a leaf so a new shoot will sprout there.

Good Luck