The only two mentions of the Greyhound are these:-
Northampton Mercury - Saturday 19 November 1864
A Newly-erected Messuage, or BEER HOUSE called the "Greyhound," situate in Alma-street, Far Cotton, containing parlour, sitting-room, back kitchen with room over, and three bed-rooms; yard, with with pump and well of water; large club-room, carpenter's shop, stable, and outbuildings adjoining, now in the occupation of Mr. George Smith. This Lot is let on a lease to Messrs. Rogers and Co., Brewers, for a term of Three Years from the 6th day of January, 1861, at the annual rent of £30.
Again in 1866 A beer house known as the Greyhound, with shop adjoining situate in Alma Street
It was sold again in September 1871 and still called the Greyhound. Not sure if it then became the Engineer or is a different building. It is definitely not the "Rose and Crown". It was at the time it was sold late in the occupation of Messrs Rogers and Co. Brewers.