Ok, I added a line to clear the transposed data. When you want to add another column of data, right-click the empty column between original and transposed data (so column D for the first extra data column) and insert a column (to keep an empty column between original and transposed data), enter your data and run the code again to update (actually create new) the transposed data.
Here is the updated code:
Dim lRow, lCol, x As Integer
lRow = Range("A1").End(xlDown).Row
lCol = Range("A1").End(xlToRight).Column
Cells(1, lCol + 2).CurrentRegion.ClearContents
For Each cell In Range(Cells(1, 1), Cells(1, lCol))
x = x + 1
Range(Cells(1, cell.Column), Cells(lRow, cell.Column)).Copy
Cells(x, lCol + 2).PasteSpecial Transpose:=True
Application.CutCopyMode = False
PS: When you remove a column of data, you will want to run the code twice to get correct transposed data.