Although with most TLS implementations they will see the size of your transfers, thus still giving away which URL you visited....

Idea: HTTP server module that appends a minimal amount of hidden random garbage data (as a comment or similar) to each response so no two responses have the same fingerprint.

user btown answered that elsewhere here in this thread: "https://github.com/technion/mod_randpad (an Nginx module that injects padding as a comment into returned HTML documents) is likely the correct layer to add random padding; the performance hit would be negligible."

If this is the first (and only) time someone visited the domain around the time of a leak, it may be a little suspicious.

Then use public WiFi.