Returning 1 Max Value when Some Counts Are Equal in Access
When you join this result set (as a subquery?) back to the main table, why don't you do another GROUP BY and get the MAX(Start)?

Select MaxCount.DOW As DayNumberOfWeek, Max(Date) As LatestDate, MaxCount.Max
From
  (
  SELECT DOW, MAX(Counts) AS Max
  FROM
    (
    SELECT DOW, Start, Count(*) AS Counts
    FROM Outbound
    GROUP BY DOW, Start
    )
  GROUP BY DOW
  ) As MaxCount
  Inner Join 
    (
    SELECT DOW, Start, Count(*) As Counts
    FROM Outbound
    GROUP BY DOW, Start
    ) As DailyCount On MaxCount.DOW = DailyCount.DOW And MaxCount.Max = DailyCount.Counts
Group By MaxCount.DOW, MaxCount.Max;


Apologies if I made some syntax errors, I don't have Access handy right now.