I always liked H. Beam Piper's approach, so let's use it:
Seconds are such a fundamental unit that we are not likely to replace them. Granted they were once based on rotations of the Earth, but they are now firmly grounded on wavelength and frequency. With the rest of the SI units, they relate to too much in engineering to be replaced at a whim.
Since day, month and year are so firmly connected to Earth, they will probably not be used between planets. Instead, most people use hours when referring to standardized events and durations. Days and years are a purely local affair having fallen out of technical terminology entirely.
Finance might be an exception; however I expect contracts will become even more baroque and obfuscated than they are now. Under the hood, financial system will probably end up working in hours.
For planetary time, I expect clocks will simply be programmed prior to arrival to reset to zero at local (or zone) midnight. Passenger spacecraft on long (and they will all be long) voyages would probably slowly shift time to the cycle of the planned port of disembarkation; so no jet lag for the interstellar traveler.