Huh?
Again, you submit LORs with your secondaries. So if you submit June 1st and shadow a DO afterwards.... your primary will still get verified and sent to schools. Then the schools will still send you secondaries. You send your LOR when you submit your secondary.
So timewise... submit in June, get secondaries around July/August. So you have until then to get your DO LOR. If you get one before July/August, that's fine... Interfolio will store it for you (assuming you're using Interfolio).