As momentum is conserved for a free particle, it is not obvious to me that the momentum becomes more certain for a Gaussian wave packet. Neither is it obvious to me that an initially Gaussian wave packet will remain Gaussian.
EDIT (02/11/2018): According to http://demonstrations.wolfram.com/EvolutionOfAGaussianWavePacket/ , an initially Gaussian wave packet can remain Gaussian (at least in some cases), but the momentum uncertainty does not depend on time.