How do I trigger a macro to run after a new mail is received in Outlook?

Try something like this inside ThisOutlookSession:

Private Sub Application_NewMail()
    Call Your_main_macro
End Sub

My outlook vba just fired when I received an email and had that application event open.

Edit: I just tested a hello world msg box and it ran after being called in the application_newmail event when an email was received.

This code will add an event listener to the default local Inbox, then take some action on incoming emails. You need to add that action in the code below.

Private WithEvents Items As Outlook.Items 
Private Sub Application_Startup() 
  Dim olApp As Outlook.Application 
  Dim objNS As Outlook.NameSpace 
  Set olApp = Outlook.Application 
  Set objNS = olApp.GetNamespace("MAPI") 
  ' default local Inbox
  Set Items = objNS.GetDefaultFolder(olFolderInbox).Items 
End Sub
Private Sub Items_ItemAdd(ByVal item As Object) 

  On Error Goto ErrorHandler 
  Dim Msg As Outlook.MailItem 
  If TypeName(item) = "MailItem" Then
    Set Msg = item 
    ' ******************
    ' do something here
    ' ******************
  End If
ProgramExit: 
  Exit Sub
ErrorHandler: 
  MsgBox Err.Number & " - " & Err.Description 
  Resume ProgramExit 
End Sub

After pasting the code in ThisOutlookSession module, you must restart Outlook.