Others may be able to provide a better solution, I'm not sure that it can be done if you want to keep the due date/time.
The preferences (cmd-comma) has a drop down to set the 'due soon' value but the smallest value is 24hrs.
The 'Due' view you mention is, I suspect, the built in 'Due' perspective (ctrl-cmd-P to bring up the perspectives window) and this filters by the 'due soon' value.
Many people only use the Due date/time field for things that affect life, cash or liberty (like submitting the tax return on time). I was tying myself in knots and had lots of 'distracting' red ink until I realised that setting due dates as reminders was the wrong way to go - I use flags now for "Today's Stuff" reserving the due field for the Critical stuff only.
It is the due date/time that is triggering the appearance in the 'due soon'
- open the perspectives window and select the due perspective.
- click on the gear at the bottom of the perspective window sidebar and select 'save window as --> new perspective'
- give the new perspective a name e.g. 'Flagged and Available'
- now edit the settings (bottom right box) so that Availability = Available and Status = Flagged.
- You might wish to check 'open in a new window' - I do this so I can use a link in the notes to pull up some perspectives that I check daily like 'Waiting For' without losing the current view.
Now go and edit your task (or create a test one) that is flagged, with a start time 1 minute away and no due date. Select the 'Flagged and Available' perspective - the task wont be there.
Wait a minute (you might need to refresh the perspective) and the task will appear.
Perspectives are the third leg of the tripod that makes OmniFocus truly useful (projects and contexts being the other two) - they are incredibly powerful but can be confusing to set up in the first place (the support ninjas can help) They are covered well in Kourosh Dini's book and also in various places on the web.