Find the names of big local distribution centres in your area and approach them directly. This could simply be driving around local industrial areas and making a note of all the distribution companies. The names will often be plastered on the sides of all their lorries.
My first job was through an agency, but the distribution centre that I was sent to said they would have taken me on directly if I had approached them first without going through the agency. Apparently, if you go through an agency the end client can’t take you on or the agency will sue them for poaching their staff/agency drivers.