Can you use something other than sodium chloride? Sodium bromide seems to be much more soluble in methanol:
Sodium chloride = 14.9 g/kg
Sodium bromide = 168 g/kg
A similar trend would be expected with other alcoholic solvents such as glycerin.
You may also consider lithium chloride. Lithium salts are more soluble in many polar solvents than heavier ...