you could always do something like have a supercapacitor which is also charged all day from the sun, and once the voltage drops below some level you begin discharging the supercap.
would take some design to boost it to the correct voltage (if you need 3 or 5V) and make sure the discharge was slow enough to last during the dark hours,
but there should be some value of supercap that would work for you if the sleep mode isn't an option. (or a smaller one if sleep is an option).