I think in the East Riding there are no leagues appointing assistant referees below the Northern Counties East Football League (Step 5/6). This league normally takes recommendations from your county FA for assistant referees, so you could contact your RDO about that.
For AR appointments below NCEFL, but requiring more travel from your location, would be the top divisions in the North Riding Football League, Lincolnshire Football League, West Yorkshire Association Football League and Sheffield & Hallamshire County Senior League.