In order to upgrade to a newer version of PHP in CentOS 6, you'll need to enable some external repositories, EPEL and Remi. You can download the repositories' information and enable them with:

wget http://dl.fedoraproject.org/pub/epel/6/i386/epel-release-6-8.noarch.rpm
wget http://rpms.famillecollet.com/enterprise/remi-release-6.rpm
rpm -Uvh remi-release-6*.rpm epel-release-6*.rpm

You can then upgrade PHP with:

yum upgrade --enablerepo=remi php

@asb YOU SAVED MY LIFE IN LESS THAN 3 MIN.

Thankfully updating my PHP version was as simple as running you suggested commands!

Thank you very much for your help.
I can confirm this works :)