There are some post war obstacles to this plan. I have taken the map you linked to and superimposed it over the map I made with all the 4th edition US targets. The result can be gotten here:
https://1drv.ms/i/s!Al7j9i_hmlNFpnTHxYlGsjogoP84
The high bandwidth telephone and television links get severed many major cities. There are a number of the telephone only links that survive, but they are still large regional areas and I have not found a single link that connects East and West coast. I did not include the lines that were listed in the legend as planned or under construction. If we were to assume that AT&T was a member of the Counsel of Tomorrow, then perhaps these planned and under construction links were completed using an accelerated timeline as the war approached, since that is known to the Counsel. They may have even put in additional links to add redundancy and work around the major choke points. This could mitigate the problem and give you the genesis of Morrownet.
But all this metal and wire just sitting on a pole or mast would quickly be scavenged in the first decade after the war if it is not near any population center that worked to preserve it. In my campaign, I have two college campuses that have a microwave relay network connecting their campuses. Granted the link is reasonable short and only comprises towers along a trade route and is a recent construction. This map shows 14 exiting AT&T towers along a route close to the trade route I made. So I don't think that is it impossible, just unlikely that any long routes would have survived.