How to save blocked site in Wayback Machine

Why isn't the site I'm looking for in the archive?

Some sites may not be included because the automated crawlers were unaware of their existence at the time of the crawl. It's also possible that some sites were not archived because they were password protected, blocked by robots.txt, or otherwise inaccessible to our automated systems. Siteowners might have also requested that their sites be excluded from the Wayback Machine. When this has occurred, you will see a "blocked site error" message. When a site is excluded because of robots.txt you will see a "robots.txt query exclusion error" message.

The exclusions are at the choice of siteowners or due to take-down notices. I'm afraid if you are asking whether you can decide to include what is presently excluded the answer is that you cannot (for other than your own site/s).

What you want -- "only show content collected after this date", is not supported by the Internet Archive's software.