I always walk between E and C when arriving on a domestic flight. An hour should be more than sufficient. E is on the same side as the C-South complex, which includes gates C29-45, and walking is pretty much the only option here. However, if you are connecting from E to the C-North concourse, which has gates C14-27, you can walk to the center of Terminal E, up through the walkway toward the Terminal D, and take the TerminaLink to the C station.

If you are arriving on an international flight (which I suspect, otherwise your E gate assignment is definitely subject to change), I would recommend taking the TerminaLink train regardless of the location of your gate in Terminal C. You exit the international arrivals area in between Terminals D/E, very close to the entrance to the train, so your walking distance is greatly reduced by doing this.

It's also worth noting that the instructions regarding transfers between CO's operations in Terminal A/B above date from 2006 and are no longer in effect. Presently, CO only runs a bus between B84 and A2, and another route between A2 and A20 on the A-South concourse to connect to UA/US operations. Therefore, a connection between A <-> C requires a ride on the TerminaLink to/from C, and a walk between the bus stop at B84 and the train station. The extra leg adds another 10-15 minutes to any connection between A/C, even for a fast-moving traveler.