Some type of joule thief for batteries? No, there's probably not enough place to stuff a step up module in there, plus it might cause undesired operation.
There are some specific rechargeable batteries with zn that have a higher voltage, 1.6v (and more when charged up), but then there's the risk of having too much voltage and causing something to overheat. There might be other technologies that provide 1.5 in the desired package.
Other than this, if there's an internal MCU that chooses the modes (and toggles between them) I'm not sure there's much that can be done.
It does sound like when the battery is low (probably low voltage like you mentioned), the lamp doesn't want to go into high mode. It likely considers the battery to be low at the voltage of the rechargeable batteries.