Hi Allesleylived, welcome to the forum
As far as I can see on the map, there are no lanes, alleys or yards called 'Pig' in that area, at that time. 'Alley' is an unlikely name for Coventry but there may have been a small stretch of road with either 'Lane' or 'Yard' that was too short to record on the map. At that time they tended to be named after the business that fronted them or had done in the past. There were no pubs called Pig that were recorded, so it would have to be another type of business. There were a lot of butchers and slaughterers in the area so it's not impossible that there was such a route.
The only thing that might fit, is an area between Cook Street and Silver Street that was called Pig Market but that is north of the Burges and what was to become Trinity Street (that didn't exist then).
There may be a reference in the newspapers or directories at the time?