I have an ancient flip-phone that I only use when travelling, to let my girlfriend know what time to pick me up at the train station or whatever. Otherwise, I just use our land line.
One of these days, though, I'm going to have to break down and check out one of these newfangled "smartphones."
Getting back to TREK, I suppose it's possible that Kirk peeks at his communicator whenever he wants to check the time, but I can't say I've ever seen an actor on TOS actually do this. And Kirk usually isn't carrying a tricorder or an electronic clipboard or whatever. (PADDs are not a TOS thing.)
Maybe it's the Yeoman's job to tell him what time it is?