For what it's worth, it's also incredibly unusual to say "Hi, have a good day" unless you are walking past someone and only see them for a few seconds. "Have a good day" is something you say as a goodbye. This whole phrase is odd, in both English and Hungarian. Maybe it should be removed ...
I have noticed that people in Budapest (at least) like to pile up the greetings. I've often heard things like "szia, hallo, jó napot" upon entering a café and "szia, viszlát, hallo" as I leave.
(I find "hallo" upon leaving to be particularly weird, especially as in Polish such an utterance could mean "Hey wait! (you forgot something!)")
Let's recognize that 'Szia, jó napot!' is a welcome greeting, not a farewell greeting - so 'Szia' means 'Hi'/'Hello' here, not 'Bye'/'Goodbye'.
When 'Szia' used in a farewell sense, it rather means 'Bye' than 'Goodbye', on grounds that both 'Szia' and 'Bye' are informal and abbreviations of another greeting.
What a strange combination of greetings! I try to imagine a situation when you could use it fellows. :-D So you are walking on the street and meet a close friend and her/his older companion or someone you don't know. So you greet them like this Szia! (to your friend ) Jó napot! (to the person who's older or you don't know). Actually I would say Jó napot! first to be polite. ;-)