Upvote:2
Your first Schengen will expire at 23:59 on 30 Sept. Your second will begin at 24:00 on 1 Oct. They are the same instant and you are legal throughout. You can use the Schengen calculator to determine if 1 Oct counts as a day of leave or as an 'arrival' day and how that will affect your overall spent leave in the Schengen zone. But for your question: yes, you are ok.