I’d suggest that this is a management problem and not a technology problem.
The way I handle escalation when I install systems (not the “right” way, just another way) would be to have each site have a Ring Group at each store (assuming there are multiple extensions at the store).
That ring group rings for 20 seconds. When that fails, drop onto a second Ring Group that has the extensions for the Call Center included. Let that ring for 20 seconds.
At the end of that, drop the caller into an apology queue. “We’re sorry - we prefer to have a person answer your call, but everyone is really busy right now. If you’d like to hold on, we will get to you as soon as we can. If that’s not possible …” and at that point you can either drop them into voice mail or initiate an automated callback when you have a rep in the call center available.
There are two problems with the “hot potato” method:
Transferring the call back and forth is just “ringing” for the customer. I know I won’t sit on the phone waiting for someone to answer for a full minute any more, and most people I know that use phones are even less patient than I am.
Transferring from the “gathering” has the problem of “how do I get back to the original store?”. @sorvani’s idea isn’t bad, but you have to be sure you are doing something that actually transfers the call back to the original store. That can be a little tricky, especially since we know the phones at the store are “unresponsive”.
Another problem/issue is what happens when the phones are all actually busy? You want to call to go straight to the Call Center. If they can’t handle the calls, the problem is going to just get worst - there’s going to be a non-zero chance that the phones are all still busy at the original store, and you are right back on the ring group. If the phones in the call center are also all busy, your call will bounce back and forth incredibly fast and then just hang up (when the call transfer depth reaches 6 transfers, IIRC).
In general, I recommend against Ring Groups in a business environment, especially in a scenario like this. I prefer the management control of a queue. Setting up a queue per store with the Call Center being in the mix (through Ring Strategies, for example) means that calls that aren’t getting picked up at the stores are going to your call center. When they get busy, there are still resources available in the queue (through additional handling legs) that get the caller to a place where they are being helped ASAP.