Roger,
Much depends upon the bullet and the hardness of the bullet, as that will dictate pressures developed, to a degree, with softer bullets spiking pressures more quickly than harder ones. Also note that most of these guns, while having a .357"-.376" groove diameter, have huge throats in them, and usually a .377" diameter bullet shoots best in them.
The loads listed below use Beartooth's BHN 21 GC bullets.
In the .375 Winchester:
BTB .377"-250g LFNGC/30.0g AA-1680/WLRP/Win Brass/2051 fps/20" Bbl. (very uniform ballistics)
BTB .377"-250g LFNGC/35.5g RL-7/WLRP/Win Brass/2210 fps/20" Bbl. (a potent load. Reduce 5% and work up)
These loads have proven very accurate and effective in both the Marlin 375 and Winchester 94BB in .375 Winchester, but have not tested them in a Contender Carbine barrel.
Hope this helps!
God Bless,
Marshall