Script to find column header doesn't work if it is second to last or last column.

Sub columnfind()
Cells(1, 1).Select
Do Until ActiveCell.Value = "My Column Header"

   If ActiveCell.Column > ActiveSheet.UsedRange.Columns.Count Then Exit Do
   ActiveCell.Offset(, 1).Select
Loop

End Sub

This works, UNLESS "My Column Header" is the last column, or second to last column, then it jumps left two columns, instead of landing on the correct column.

The purpose of this script is to select a cell directly in that column that I was searching for.