Figure out a little time window without bruteforcing

I have a website which provides me with a simple printout most of the day. There is a very small time window (1 minute) each day where the website returns the proper response. I would like to know if there is a smarter way of finding this small time window instead of bruteforcing it.

This is what I have tried so far:

1. I have checked the request header and it does not include a timestamp => The server checks its local time => Cannot manipulate this.

2. The website is up 24/7 => Cannot use an online page to check when the website is up.

3. The html file of the current page only contains the print-out

   < html > < head >< / head > < body >not the right time< / body> < / html >

=> I assume there must me a second html file for the response within the small time window. Therefore, I had a look at the overlying folder, but the index in there is not useful & I did not find any other files in there.

4. Tried to manipulate the url, but that was not successful either.