We show that the exact orientation distribution function (ODF) of the Onsager model for the nematic liquid crystalline phase of thin hard rod-like particles at high density can be scaled onto a single curve, which needs to be determined numerically only once. The scaling solution is in better agreement with simulation results on a system of sphero-cylinders of length-to-width ratio L/D = 40 than the Gaussian ODF, hitherto believed to be the best representation. The phase behaviour determined from the scaling solution, and from its first-order low density correction, are compared to the exact results.