Iris, "have a nice day" and these types of phrases only make sense when there are many hours left in the day. If you are going to sleep, you wouldn't say this to someone who lives in the same time zone. You would say "good night". But if you are Skyping with or writing an email to someone in a time zone that is 12 hours different than yours, it makes sense.
In that case, you can say "enjoy your day" or "have a great day". Instead of "great" you can say "wonderful" or " awesome", and so on.