Add a boost regulator to take supercap and maintain 3.3V output.
You touched briefly on "Linear regulator, Don't even think about it. Any Power used to heat the regulator won't be available to move anything else... There are 3 or 4 terminal switchers that are either boost, buck and or variable for both topologies as well as boost-buck which is neat because Vsupply can swing about the output voltage. (not as efficient but still 80%+) where a linear has to be specially fed to get 75%. Lest we forget there are a lot of "Low Dropout" regulators that have an ugly fact buried on page 10 of the data sheet and that is the required internal Control Current doesn't flow in the load like an NPN device does (emitter Follower) but to ground as the only way I know to do it is with a P type device (low dropout). It is arguable that P type "Fets are available... Now that technology took years to get to regulators, still not used in the greatest majority of last gen regs so there is a price carefully hidden and that is that the control circuits need lots more current for a given load, compared to no load. Literally I use linears where low noise and isolation are required as they are easier to bypass that a switcher but only required when sub millivolt noise is required @ milliamp supply currents or there is a radio involved of any kind. Blocking is just as good as a poor antenna as far as "Not" receiving a weak signal. Or other low noise, small signal, analog requirements. If there is a choice at all use a switcher. IMO and experience in designing low power battery operated solar charged devices.