I live in Surrey and use a Garmin Montana, but I also carry maps when I go trail riding as a backup in case the Montana fails. Also, if I need to change my route - e.g. if a lot of time is lost due to breakdown or accident - I find it easier to plan a revised route by looking at large maps rather than a small satnav screen.
I buy my maps from Dash4It, if you were to buy OS Explorer maps 145 & 146 you'd find enough lanes to cover a winter day's trail riding in Surrey, and it would cost you less than £12 delivered
http://dash4it.co.uk/catalog/product/view/sku/OSE145Once you've got the maps, you could mark them up by reference to
http://www.qwerf.com/greenlaning/?county=surrey or better still, join the TRF and go to the Surrey group's monthly meeting in Ripley, where they will have a set of marked-up maps for you to reference, further details here
https://forum.surreytrf.org.uk/viewtopi ... c962fbdc33Whilst I prefer to use a dedicated satnav rather than a smartphone, many riders use the Viewranger app on their phones. If you search this forum for 'Viewranger' you'll find loads of useful threads.
Hope that helps