I’m running a new dryer line upstairs with 10/3. I bought 50’ hoping it was enough but alas, I’m short about 3’. I have 20’ of 10/2 leftover from a different project and wondered if I could cut a splice of black wire and slide it into the the 10/2 to effectively make it 10/3. I haven’t tried yet and realize the sheathing may not want to cooperate, but since it’s not too far thought it may work. Is there any reason why this would be unsafe?
No, that's not allowed. NEC 300.3.
If you have a location that will remain accessible, you can install a junction box - I would use a 4x4 steel box with cable clamps made for 10/3 - and then do your splice there.
If you haven't cut or installed the wire, you can return it to the big-box store, and then get the right thing for your job.
If your socket is labeled for aluminum wire, you can use #8 or #6 4-wire aluminum, which will take some pain out of the price of wire. There is nothing wrong with aluminum wire at these large sizes.
Note that all fasteners which state a screw torque need to have that torque set with a wrench. That's in Code now, at 110.14. That is because mis-torqued screws are a regular point of failure.
Lastly if you are stuck buying "small spools of wire", that's a procurement problem from buying at big-box stores. Better lumberyards and hardware stores will sell you wire by the foot.
As already answered, that would be against code. As to why, the answer, as is often the case, is heat. Any current running through wire produces heat. Multiple wires stuck inside a cable sheath is even worse, which is why at certain sizes the same size wire is rated for more current if loose in a conduit than if inside a cable. Your idea is to stuff another wire inside a cable, which if it would fit (it won't) would make the heat problem even worse. Even in conduit, even if there is plenty of extra space, extra wires (multiple circuits) lower the current carrying capacity (according to code) because of heat.