That suggests you're not getting enough pitch range on the tail rotor, or as you mentioned, turning too slow. Difficulties with lift suggest perhaps the latter?

Quick checks for pitch range: if you have a computer radio (don't remember, sorry) make sure the end points on the rudder are set to their full travel. If your gyro has limit settings, then turn them up some. You're looking for a setting that will get the servo to reach the limit of the tail travel at full deflection. For example, with the GY401, I'll put the gyro in rate mode and then the rudder full over, then turn up the limit pot until the gyro binds. If the servo doesn't bind in the other direction, then I'll move the servo arm or lengthen/shorten the pushrod until I can get the travel centered between the two, then adjust the limit pot down until they don't bind in either direction. It's a bit of a balance to get to.

Once the servo travel is centered across the limits of the tail, then you'll probably have to retrim in flight on the tx. There is, again, a balance there, and you'll want to adjust the mechanical trim again to get the zero trim center on the servo close to a zero yaw in flight, but bias your adjustments toward right pitch on the tail rotor. As you've found, you don't need much left travel to get left yaw because of the torque, so you want to have as much right pitch available as you can get. You'll find, though, that if you have the servo electronically centered at a point that the heli doesn't yaw left, that at center rudder on the tx the tail rotor is pitched right. Then you'll have all the pitch right available to you for right yaw.

HTH