We've got a couple of cordless Radio Shack phones that will do the same thing once in a while. I'm not sure what we're doing to cause it.
It seems like if I make a call, make a mistake dialing, hang up real quick, get dial tone again, and redial. The call goes through OK and then when I hang up, the phone will ring. I pick up and I'll usually get a recording telling me that "if i want to make a call, hang up and try again."
It's almost as if when I hung up and redialed, the first attempt got put on hold, then after completing my call on the second try and hanging up, the phone rings reminding me I have a call on hold.
If I call the local library's story line, start listening to the story, hang up, get dial tone again, dial my number, and get a busy signal as expected, then hang up, the phone will ring. I pick up, and there's still the story being told.
It almost sounds a little bit like the way 3 way calling works, but as far as I know we don't have it.

Yes, you're right in what you said at the end of your post; this is coming from 3-way calling. Many phone companies have now enabled 3-way calling and all the other "custom calling features" on a pay-per-use basis, so your line probably does have 3-way calling enabled on it.

This will not occur on all calls because if it is on your local exchange that you have gotten a busy signal, the phone company's computer is "smart" enough to know that you would not want to put a busy signal on hold and dial a second number. However, if the busy signal is on a long-distance number, your local phone company's computer doesn't know it's a busy signal, so it will let you put it on hold and dial a second number.

You need to keep the phone hung up longer before making a second call. In some cases, you may be charged long distance for the call that's hearing a busy signal but is on hold, and in others you might be charged for a use of 3-way calling if it's on your line as a pay-per-use feature.

In certain situations, call waiting can also cause this. If you make a call and then abandon it to take a second call that beeps in on call waiting, often times you do not properly disconnect the first call, and this will cause your phone to ring when you finish the second call and hang up.

This can often happen if you were on hold at some type of "customer service" number or similar which is a toll-free long distance call. If the first call (the one you abandoned) was a local call, this shouldn't happen as often because your local phone company's exchange can tell that the first call was terminated.

If you are on a call that you want to abandon to take a second call that beeps in on call waiting, do not just flash the switchhook to click over. Hold down the switchhook or hang up the phone and wait for it to ring. This will properly disconnect the first call.

My issue is with a Cordless ATT Hand Held Phone. After I finish talking and hang up. My phone Rings Back.

My mother's cordless phone of some Uniden model or other sometimes does that. Probably caused by something wonky with the local (independent) landline telephone company in her town. I don't know if it happens on the wired phones she has in a couple bedrooms, they're rarely used.