Whether a driver circuit that powers the LED in a torch can handle more than 4.2v (the fully charged voltage of a LiPo) is dependent on how it drives the LED, and the voltage tolerance of the components in the driver.
We understand that LEDs have a very narrow voltage band within which they will draw a reasonable amount of current. Increase that voltage slightly, say 10% above what it likes, and suddenly you go from reasonable operation, to drawing Crazy amounts of current and the LED burning up almost instantly. The ideal voltage for an LED is usually around the 3.2-3.5v range (not sure on exact figures, but this is where it generally is) and is somewhat determined by the energy required to move an excite an electron sufficiently to release a photon of light of the blue wavelength which we then convert to white light through phosphors.
Connecting a LED directly to a battery is only done in carefully managed cases because the LED generally draws almost too much power to survive normally. We use drivers to keep the voltage a little lower to keep things safe. A driver takes your input voltage, and presents the ‘ideal’ voltage to the LED. How it does it is the answer to your question of whether your 5-6V source will work.
Fancy drivers that have buck (AND) boost circuits are able to handle any input voltage, and change it into a lower (or higher) voltage that the LED wants, and the LED will only ever be exposed to a safe voltage. These drivers can be designed to handle one, two or more cells, and are generally stated as having compatibility to 8.4v, or 12.6V (or similar). These drivers you would be able to use, as simply supply it a voltage within what it can handle and your fine. They will only give the LED a voltage that is generally safe for most LEDs.
Many cheaper LED torches use Linear drivers and are fundamentally connecting the LED directly to the battery. For brightness control, its simply connecting power to the LED on and off really quickly to create a lower “averaged” voltage. The LED and driver need to be able to bear the effects of the full supply voltage. In a single cell light, this is at most going to be 4.0V or so even with a really top notch cell, and your going to be drawing a high amount of current for a short period. Given that its maybe only 0.5-1.0V above what the LED wants, its generally doable. But if you increase your voltage to 5-6 volts, thats HUGELY outside of what the LED itself can handle and you get Instant POOF!
This is assuming the components in the circuit are able to handle a higher voltage too (which is likely, but not guaranteed)
In short, cheaper linear drivers - No Go!
Switching drivers that have a wider input range, - Most likely Yes!