Copy table to the first unused row on another sheet

Try:

Sub CopyRange()
    Application.ScreenUpdating = False
    Sheets("Today's Queue").UsedRange.Offset(1, 0).Copy
    Sheets("Queue History").Cells(Rows.Count, "A").End(xlUp).Offset(1, 0).PasteSpecial xlPasteValues
    Sheets("Today's Queue").UsedRange.Offset(1, 0).ClearContents
    Application.CutCopyMode = False
    Application.ScreenUpdating = True
End Sub