You've given one possible reason to the moss in your post
...as we are surrounded by forest on three sides...
I admit it will depend what sides of the field the trees are growing on but the trees will be reducing the amount of light that comes into the field (to a certain extent). The field may not get as much sun and through wind in order to dry out the ground and so as a result will leave a damper than normal area leading to moss growth.
Best solution would be an application of ferrous sulphate (moss killer).
You could try and improve the drainage/air levels in the field - aerate the sward to let air into the sward and surface of the field to help it dry. You could then harrow the field before applying more grass seed and rolling.
Moving on after the grass has re-established - I'd harrow the field every year and aerate it to.