How can I add another action with the button

It's not clear why you "can't" add these steps but as I intimated in my comment you already have the code to do that.

Having looked at this again, you may be having problems because your code entirely relies on having the correct sheet active in the workbook when it runs. Always explicitly state which sheet you are referring to and that should avoid the problem e.g.

Sub SalvaFile()
    Dim wkA As Worksheet, ur As Long, j As Long, riga As Long, nomeFile As String
    
    Dim wkB As Worksheet: Set wkB = Worksheets("Sheet1") 'change this to your source worksheet
    
    Dim percorso As String
    Set wkA = Worksheets("Scadenziario")
    wkA.Range("A2:T" & Rows.Count).ClearContents
    ur = wkB.Range("A" & Rows.Count).End(xlUp).Row
    riga = 2
    percorso = "C:\Users\Itaerxga\Desktop\Scadenziari\"
    Application.ScreenUpdating = False
    Application.DisplayAlerts = False
    On Error GoTo errori
    For j = 2 To ur
        wkB.Cells(j, 1).EntireRow.Copy wkA.Cells(riga, 1)
        riga = riga + 1
        If wkB.Cells(j, 5) <> wkB.Cells(j + 1, 5) Then
            nomeFile = wkB.Cells(j, 5)
            wkA.Copy
            'Instead of using ActiveWorkbook use the name of the workbook!
            ActiveWorkbook.SaveAs Filename:=percorso & nomeFile & ".xlsx"
            ActiveWorkbook.Close
            wkA.Range("A2:T" & Rows.Count).ClearContents
            riga = 2
        End If
    Next j
    
    'Clear the contents after saving all the files
    wkA.Range("A2:T" & Rows.Count).ClearContents
    'select cell A2 on the Scadenziario worksheet
     <s>wkA.Range("$A$2").Select</s>'Select and Activate can cause unintended effects 
     wkA.Range("$A$2").Show
    
esci:
    Application.DisplayAlerts = True
    Application.ScreenUpdating = True
    MsgBox "Copiato"
    Exit Sub
errori:
    MsgBox Err.Number & " - " & Err.Description
    Resume esci
End Sub

You are also relying on the workbook with this code never becoming active - have a look at this sub-routinePublic

Sub Demo()
    Dim countWbs As Integer
    countWbs = Workbooks.Count

    Dim i As Integer, wkb As Workbook

    For i = countWbs To 1 Step -1
        Set wkb = Workbooks(i)
        If wkb.Name <> ThisWorkbook.Name Then
            wkb.Activate
            'Do something with the Active Work book - e.g. your save thing
            Debug.Print wkb.Name
            wkb.Close
        End If
    Next
End Sub

Things to note:
- because I am using the index to the workbooks collection I have to start at the end and work backwards (so that book number 1 is always at position 1 and is always the same workbook even after one is closed)
- the Activate is not necessary if I explicitly refer to wkb in whatever I am doing
- I make sure I don't close the workbook that contains my code!