Change your code to this:
Dim lr as Long
lr = Sheets("Sheet1").Cells(Rows.Count, 1).End(xlUp).Row
Set sourceRange = WorkSheets("Sheet1").Range("A1:A" & lr)
Set fillRange = sourceRange.Resize(, sourceRange.Columns.Count + 1)
sourceRange.AutoFill Destination:=fillRange
EDIT:
Note, as it's unclear from your question, if you also wish to detect last column,
then:
Dim lc as Long
lc = Sheets("Sheet1").Cells(1, Columns.Count).End(xlToLeft).Column
will detect it for you, you can adjust your sourcerange, and set your fillRange to be the rightmost adjacent column
Set sourceRange = WorkSheets("Sheet1").Range(Cells(1,1), Cells(lr, lc))
Set fillRange = sourceRange.Resize(, sourceRange.Columns.Count + 1)
sourceRange.AutoFill Destination:=fillRange
will do the trick!