We study the thickness of wetting layers in the binary-liquid mixture cyclohexane methanol. Far from the bulk critical point, the wetting layer thickness is independent of temperature, resulting from the competition between van der Waals and gravitational forces. Upon approaching the bulk critical temperature [t=(Tc-T)/Tc-->0], we observe that the wetting layer thickness diverges as t-β ^ with effective critical exponent β ^=0.23+/-0.06. This is characteristic of a broad, intermediate scaling regime for the crossover from van der Waals wetting to critical scaling. We predict β ^=β/3~0.11, with β the usual bulk-order parameter critical exponent, showing a small but significant difference with experiment.