You could create a separate script task that would count the files in the various subdirectories using an array list containing the subdirectories you want to search along with the System.IO.DirectoryInfo method and write the result to a global variable.
Here is the code that you could use in the script task.
Dim subdirs As New ArrayList
Dim value As String
For Each value In subdirs
Dim di As New System.IO.DirectoryInfo(value )
Dim aryFi As IO.FileInfo() = di.GetFiles("MyFileNameRoot*")
Dim Count As Integer
For Each fi In aryFi
Count += 1
Dts.Variables("Count").Value = Count
You would have to create a global variable named Count and add it to the Read/Write variables in the Script Task Editor.
To learn a enough about script tasks and global variables to make it work you could refer to my previous article Copy-and-Paste SSIS Execution Monitoring with Failure-Alert Email