My guess is that it would help if you specify a date. Then they can see how many bookings they have from members on that day, and make a decision, instead of putting your name on a spreadsheet and MAYBE getting back to you in the spring.
It would help if the date is not during their prime months [July will probably be difficult].
I don't have inside info from Sand Hills, but this approach is what the starter at The Old Course told me years ago, as a walk-up. It didn't do any good there to go over and ask if you could play later . . . he didn't know how busy the tee would be "later". But if you were ready to go right then, and there was a gap for whatever reason, off you'd go!