Have the slider show exact times

I love the functionality and design of the slider. But when I move it to either side of the Now, I'm never sure exactly where I'm setting it. "12:20ish" is the best I can do.

The thing about this tool is that it's actually ridiculously accurate. Combined with building outlines in Google maps, I can see where the sun will be in relation to various architectural features at various time. The problem is that I can't see which time (i.e. down to the minute) I've set the slider to when I've lined up a given solar/terrestrial configuration.

If there were a clock field between the date field and the "now" button, users could enter a specific time directly, or use the slider to find a certain configuration, and see from this field when it would take place.