No. My understanding is that the power must be used at the time it is bestowed, or it goes to waste. The Hunter doesn't 'lose it completely'; it means that s/he will revert to his or her normal power if the amplification is not used.
As DP has noted, - there is a difference between powers that are 'once off' and recurrent powers, and this will have an effect on how the 'amplification' power gets used.
Other than that, the powers that are normally used each night, will continue to be used - it is just that for one night, the use of these powers will be doubled if that particular player happens to be the target of the Amplification Power.
actually, the way it is written -and was designed- is in fact that the amplification FORCES the power amplified to be used immediately at its amplified strength, or it ALL goes to waste.
the point is that the amplification doesn't give the use of an additional iteration of the power, it enhances the current power potency.
so if the hunter is targeted, they have to take a decision whether to use the instakill now (doubled) or not to use it at all.
it is the equivalent of the rioter, but applied to powers. one person that is publicly rioted will have their vote doubled. they cannot avoid it, the only option the retain is who to double-vote, or to cast a no-vote. but they are not going to have a single vote that day.