Have you tried to completely wait out those 360 seconds? It’s not just posting that restarts that counter, but rather a bunch of other stuff like using the forum search. Time, as you probably know, gets reduced as your Activity increases:
See:
https://bitcointalk.org/index.php?topic=5369641.msg58369839#msg58369839
https://bitcointalk.org/index.php?topic=5356770.msg57794170#msg57794170

Note that, as mentioned above, if the recipient account does not allow for Newbie accounts to send PMs to him, you won’t be able to. The above post is complemented by this 2018 modification: https://bitcointalk.org/index.php?topic=2728840.0 (requiring people to opt-in to actually receive them).