Default date source contextual filter

I too would like to be able to control the calendar with a contextual filter.

I was trying to find the same feature, but I just found out that if you add the following param "&initialDate=2023-07-28" to the URL, the JS will handle that for you.
So, if you combine the inialDate param with an exposed filter "between", I think you can achieve what you want.
i.e.: ?field_date[min]=2023-09-01&field_date[max]=2023-09-30&initialDate=2023-09-01

Note: I'm using the SmartDate module field.