That's my point; as a subscriber, you can prefix a long distance call with a routing code to avoid, for example, a shut down long distance network without any administrator changes. Routing to the long distance networks is done independently through the local network, so if AT&T's long distance network was having issues, it'd have no impact on your ability to access Verizon's long distance network.
There's actually no technical reason why you couldn't do that with IP (4 or 6); although you'd need a approriately located host to be running a relay daemon[0].
0: ie something that takes, say, a UDP packet on port NNNN containing a whole raw IPv4 packet, throws away the wrapping, and drops the IPv4 packet onto its own network interface. This is safe - the packet must shrink by a dozen or two bytes with each retransmission - but usually not actually set up anywhere.
Edit: It probably wouldn't work for TCP though - maybe try TOR?
There are plenty of ways to do what you're describing, and they all work with TCP. Some of them only work if the encapsulated traffic is IPv6 (and a designed to give IPv6 access on ISPs that only support IPv4). Some of them may end up buffering the TCP stream and potentially generating packet boundaries at different locations than in the original TCP stream.
