The pig and cattle market was in an alley that ran from Bartholomew St, by the Falkland Arms to Cheap Street by Flint's Brewery (later the Carlton Cinema). In the 1860s & 70s the issue of clean water became a national obsession and it was recognised that the drainage in this area was very close to a fair number of wells - in short the market was a health hazard. Within the alley was a pub called the New Inn, landlord Richard Goddard. 

The pub got plenty of business from users of the market, but it also owned the pens used to hold the livestock and charged for their use. So, when the council decided to build a more sanitary market alongside an alley to the south, opening it up to become Market Street it pretty much put the New Inn out of business. To compensate the landlord and owners (Strange & Co) the prime site on the corner of the new street was allocated to them for a new pub - the New Market Inn or New Cattle Market Inn or just Cattle Market Inn (this last one being pretty confusing since the old New Inn was also called this at times). 

Goddard left the New Inn and took over the new pub in 1872, the licence of the old pub being transferred with him. Plenty's bought the old market area and expanded into it, blocking off the alley. From 1905 and the opening of the racecourse Newbury became associated with horse racing and the New Market became Newmarket in the minds of many.