While you can't change the default time for 'do not deliver before', you can use VBA to set the delivery time and save a few steps.

If you prefer to use a utility to delay delivery, utilities at listed at Tools to delay sending a message in Outlook.

In the code example at Create a deferred Birthday message for an Outlook Contact I use the contact's birthday as the new deferred to date.

In the code samples on this page, the date is calculated to be today at 8:24 AM. By changing the decimal, you can send it at other times: .25 = 6 AM, .50 = noon, .75 = 6 PM.

You can also use a specific time by entering it in this format:

SendAt = DateSerial(Year(Now), Month(Now), Day(Now)) + #8:00:00 AM#

The first two samples on this page allow you to defer only the messages you choose, by opening a new message form using the macro that sets the deferment, with the first sample opening a new message form with the deferred time. The second sample is for users who pick from the contacts folders - select a contact then run the macro to open a new message addressed to the contact.

To "defer all messages you send by up to 120 minutes", you can use a rule. If you want to delay all messages sent between specific times (such as outside of normal working hours), you can use an ItemSend macro. See "Delay Delivery of Messages Sent at Specific Times" for a sample macro.

To enter a time each time you create a message using the macro, replace the SendAt = line with the following line. This will open a dialog for you to type the delivery time. Cancelling the dialog will open a message form without deferring delivery.

SendAt = InputBox("Enter a time or click Cancel if you don't want to defer this message. ", "Send this message at", Now())

Add the macro to Outlook's VB Editor then customize the ribbon or toolbar, assigning the macro to a button. When you want to send a deferred message, use the 'Deferred' button you created.

Defer message macro

Public Sub SendDeferredMessage()
Dim objMsg As MailItem
Dim SendAt

Set objMsg = Application.CreateItem(olMailItem)

'send at 8:24 AM. .25 = 6 AM, .50 = noon
SendAt = DateSerial(Year(Now), Month(Now), Day(Now)) + 0.35

  objMsg.DeferredDeliveryTime = SendAt

  'displays the message form
  objMsg.Display
  
  Set objMsg = Nothing

End Sub

Address message to selected contact and defer

Public Sub SendDeferredMessagetoContact()

Dim SendAt
If TypeName(ActiveExplorer.Selection.Item(1)) = "ContactItem" Then
 Set oContact = ActiveExplorer.Selection.Item(1)
 
'send at 8:24 AM. .25 = 6 AM, .50 = noon
SendAt = DateSerial(Year(Now), Month(Now), Day(Now)) + 0.35


Dim objMsg As MailItem
Set objMsg = Application.CreateItem(olMailItem)

  objMsg.To = oContact.Email1Address
  objMsg.DeferredDeliveryTime = SendAt

  'displays the message form
  objMsg.Display
  
  Set objMsg = Nothing

Else
MsgBox "Sorry, you need to select a contact"
End If

End Sub

Use a utility

If you prefer not to use a macro, there are a number of utilities available that you can use for delaying or scheduling messages.

They are listed at To delay sending a message in Outlook