Nobody "walked into" a HEO job because they had a degree. You might not have been able to apply for the role without a degree (although I've not seen that very often tbh) but they still had to go through the same process as everyone else.
Are you willing to say what dept the customer contact centre is in. They are each very different and that might influence your decision. However, most call centre jobs are tough.
Are the set hours permanent or just during training or just because you're temporary?
If its a temporary role, I think you are still able to access the internal jobs and in my experience most people are more than happy to support you to get a permanent job/progress as long as you are good at your job and pleasant to work with.
Also, in my dept, most temporary or fixed term roles do become permanent. Obviously there is no guarantee of that.
If you don't have a job right now or are on a lower wage or don't like your current job, I wonder what there is to lose? Its not like you cant keep applying for other jobs.
Also, and this is often just down to luck and I guess dept specific but there are sometimes the opportunities to gain additional qualifications eg I know someone in a different dept who although he has worked there for 10 years is being supported through a paid apprenticeship to get an NVQ (or maybe A Levels? Some sort of recognised qualification anyway).