The 308 is a brilliant rifle but can anyone add what would be a good move forward. I was thinking a 7.62 but I have no experience with this kind of caliber.
yes you do, your 308 is a 7.62mm round, almost anatomically identical to 7.62x51 NATO and so close that you can shoot 7.62x51 out of a 308(do not try in reverse order).
30-06 is a small step up but is a little better suited to long range hunting. if I was going for a full step up, as in looking for flatter shooting with a lot more oomph, I would go with a 300 magnum of some sort. 300 winchester magnum and 300 weatherby magnum are great cartridges for long range hunting but I can't vouch for availability in Scotland.