@jess, I also tried the following:
[root@player conf.d]# curl -I -v https://player.HIDDEN/api_v3/index.php
* About to connect() to player.HIDDEN port 443 (#0)
* Trying HIDDEN... connected
* Connected to player.HIDDEN (HIDDEN) port 443 (#0)
* Initializing NSS with certpath: sql:/etc/pki/nssdb
* NSS error -5978
* Closing connection #0
* Problem with the SSL CA cert (path? access rights?)
curl: (77) Problem with the SSL CA cert (path? access rights?)