When running mysql_secure_installation and entering the new password I got:
Error: Can't connect to local MySQL server through socket '/tmp/mysql.sock' (2)
I noticed when trying the following from this answer:
netstat -ln | grep mysql
It didn't return anything, and I took that to mean that there wasn't a .sock file.
So, I added the following to my my.cnf file (either in /etc/my.cnf or in my case, /usr/local/etc/my.cnf).
Under:
[mysqld]
socket=/tmp/mysql.sock
Under:
[client]
socket=/tmp/mysql.sock
This was based on this post.
Then stop/start mysql again and retried mysql_secure_installation which finally let me enter my new root password and continue with other setup preferences.