It can work, but to absolutely ensure the regulator will output a stable 5 VDC within a reasonable tolerance the input has to be at least 1.5 Volts above 5 VDC.
Edit: And 7.5 VDC can be easily achieved with standard alkaline batteries and is much more common voltage level for wall-warts to output than 6.5 VDC. Technically though, a nominally 6 VDC lead acid battery with a full or nearly full charge will have an actual voltage above 6 VDC, and probably above 6.5 VDC. So depending upon the additional current draws, a nominally 8 VDC, 10 W or more solar cell could keep a battery at a high enough voltage for a dependably stable output.