Autonomous robots have the potential to fundamentally transform conventional farming methods, e.g. by enabling economically viable farming of sloped arable land. However, navigation on slopes in harsh conditions is challenging for robots as they must be prevented from both slipping and tipping. Tethers provide one means of enabling robots to traverse steep and complex terrains with additional advantages of recoverability and connectivity for data and power transfer. Current controllers (for robot and tether) consider only the slipping condition for such robots, meaning that the tether must be mounted low on the robot. This limits the use case and can lead to entanglements and entrapment of the tether. By introducing stability criteria for tethered robots on slopes and developing a novel controller that seeks to avoid both slipping and tipping, we demonstrate how the tether mount height can be raised from 0.27 to 0.81 m on a robot, to enable, for example, the tether to be used above high crops or obstacles. This controller is demonstrated in lab and field conditions on slopes up to 0.4 radians, which translates to approximately 51% of slope. Thereby, the new controller significantly extends the practical usability of tethered robots for agricultural applications on steep slopes.