My latest info was that CentOS/RHEL 6 ships OpenSSL 1.0.1, while 7 ships OpenSSL 1.0.2.
The workarounds known for OpenSSL (removing the old root is one of them) only work on OpenSSL 1.0.2. Unless RHEL 6 has backported patches*, I'm not aware of anything you can do for those.
The CentOS/RHEL 7 discussion was here: RHEL/CentOS 7 OpenSSL client compatibility after new chain - #3 by ghen
*There were no patches for RHEL 6 back in May this year, but I don't know if the situation has changed since then.