"...get something into the contact's inbox" -- yes, this. You can avoid sitting and waiting without being rude or pushy either.
With a little creativity, there's usually a way to keep communication open without seeming rude by asking again about whatever it is you actually want to know.
Ideally, something that prompts a short reply -- ending with a simple yes/no question, for example -- so that they will actively respond to you, and feel some obligation to also give you a short update on The Big Thing.
You can also amiably pester someone repeatedly by giving them a new reason for the new request.
If you're communicating with someone very busy, also be very sure to make any email message as simple and terse as you can.
With a little creativity, there's usually a way to keep communication open without seeming rude by asking again about whatever it is you actually want to know.
Ideally, something that prompts a short reply -- ending with a simple yes/no question, for example -- so that they will actively respond to you, and feel some obligation to also give you a short update on The Big Thing.
You can also amiably pester someone repeatedly by giving them a new reason for the new request.
If you're communicating with someone very busy, also be very sure to make any email message as simple and terse as you can.