IMO it makes sense to have a couple (at least) of different pension providers/plans, in case one has a problem.
You can transfer that pension pot to another scheme/manager. You cannot just transfer it to unit trusts outside a pension without it being taxable (which doesn't make sense). You can transfer it to a pension provider which allows unit trust investments or a more flexible SIPP - Self Invested Personal Pension.
I have 3 separate schemes with 3 different providers, in general the employer picks a pension fund manager/provider and they manage it so in such a case there isn't a requirement to move your pension once you leave a job.
I assume it's a defined contribution type scheme, if it's defined benefit (e.g. final salary) do not move it