On some maps of that era the dotted square is shown as a building, but doesn't appear at all on earlier maps.
"The Limes", on the above map, is the long L-shaped building running up from the beginning of Village Road, it also had a lodge on Kings Rd/Teehey Lane (whichever it is at that point). I am fairly sure the Old Hall refers to "The Limes" as it was a very old building (as is Victoria hall just across the road), on older maps the name appears in exactly the same location as the name Old Hall on more recent maps.
We don't do charity in Germany, we pay taxes. Charity is a failure of governments' responsibilities - Henning Wehn
Just had a look at Cheshire tithes map,the Tithe map just has buildings on the site the next map it is called the Limes 1875 and the next map calls it Old hall,the British Legion is or was called Old Hall farm,the Limes appears to cover the area that Roland ave was built on,names of area's/buildings do change with time,could be the building/house called the Limes was knocked down and locals would say Old Hall for the area and it stuck,well that's my thoughts.
Found an interesting piece in the Book Fir-Bob Land by Allen Alsbury,Countryvise books ISBN 1 901231 17 8 Copyright
My interpretation on reading Pages 5-6-7 Old Hall Farm the quality of workmanship suggests it was no ordinary farmhouse no direct evidence to suggest it was the local manor house or it was the site of an older hall by the Bebington family,their is no explanation of it's rather grand building,some bits of the building on the south side of the door,it is scheduled as of special historical interest. Suggest you go to the library and try to get the book out.
HB Hall was off Kings Lane. Built in 1840. Demolished sometime after 1963. Info from Fir bob land by Allan Alsbury, chapter 9. There is a photo and location map in the book. ðŸ‘
I have some maps showing the land around Higher Bebington Hall being parcelled up for sale and development in 1958. For context, the road that has been added in pencil is Mount Drive. the Hall seems to be where Mountway is today?