Many public library systems give their patrons free access to the NYT. I was able to set this up with mine (San Mateo County Library system) in about 5 minutes from home. Another commenter mentioned incognito windows, but my understanding is that the NYT has started blocking this avenue.

Another alternative is to read via Pocket (I know, not everyone likes them), which appears to give unlimited access. Just save articles to your Pocket and open it there!