Do I understand you to mean that you would use one of the 110V-style outlets for 110V, and the other for a 24V relay control signal?
It's a really bad idea to use two connectors of the same type for different power signals. The potential for confusing them and blowing something up is too high.
If the 24V signal is low-current (less than 1A), then you could use a 1/4" ("phone") plug, such as Switchcraft 280. They're used for guitars and the like, so they are very easy to find and are inexpensive.