Can I transit via Schengen with visa-free days expired?

I have read some (or all) related questions and answers on this site, but I still have doubts whether or not I understand the situation precisely.

My situation:

Non-EU citizen, but a visa-free citizen for Schengen.
Before getting a residence permit in Sweden, I used up all my Schengen visa-free days.
My residence permit by Sweden expires on the same day on which my residence permit by France begins. Which happens on the day just before the day on which my visa-free Schengen period renews.

Does this mean that:

I have to travel directly from Sweden to France and precisely on the day on which my French permit begins? I.e., I cannot legally layover in another Schengen country?

Update: Oops, I have just noticed that my Swedish residence permit is for 92 days (3 months), while the Schengen area calculates based on days, not months. This means that my Schengen area passport time renews two days before I have to leave Sweden. This in turn means that I will be legally able to do layovers in Schengen starting 2 days before the deadline to leave Sweden :)

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Can you transit through Schengen without visa?

No, you cannot. If you have to leave the international transit area at the airport where you have landed in order to make your transfer, you cannot transit without a visa and you need a full short-stay Schengen visa.

Can I stay in Schengen Area after my visa expires?

If you apply after your visa expires, even just a day later, then you will be deported for overstaying your visa, despite of your reasons. Therefore, take care to apply at least a week before your current Schengen short-stay visa expires.

What happens if you overstay your 90 days in Europe?

The Schengen law states that you can't stay in the Area for more than 90 days. If you do, you're subject to a fine and possibly deportation and being banned from re-entering the Schengen Area.

Schengen Visa Types | Schengen Transit Visa - Transit Through the Schengen Zone by Air or Sea

Sources: Stack Exchange - This article follows the attribution requirements of Stack Exchange and is licensed under CC BY-SA 3.0.

Images: Nicola Barts, Nicola Barts, William Fortunato, William Fortunato