Miss Julie's suggestions are exactly what I would say for " If you need other information contact me"
" If you needed other information contact me" doesn't really make sense as it is in the perfect tense, but you would need to use the present tense, which is your first suggestion.
the reason you need to use "more/additional" rather than "other" is that "other" suggests the person didn't need to know what you have already told them, but "more" implies they want further information as well as what they already know
When I reply to e-mails at work I invariably end the reply with "Please call or write if you require any additional information."
I prefer phone contact so I always include the "Please call...". That is optional. (You understand that eventually humans will lose the ability to speak and will text exclusively. This is my effort to forestall that eventuality.)