Can you make it? It's possible, if all the stars are aligned (incoming flight on time or early, parked at a jetway, you're towards the front of the cabin, there's not too much queue at immigration, luggage is delivered quickly, and you get to the check-in counter quickly).
Will you make it with any kind of certainty? I wouldn't bet much on that.
With checked luggage, the main constraint is being able to check-in for your flight before the deadline, which seems to be 30 minutes before departure (Sky Express domestic check-in deadline). Which gives you one hour to deplane, get to immigration, queue, be processed, wait for your luggage, go through customs, find your way to departures and the check-in for your flight, queue, and drop your luggage. After that you'll still have to go through security and to your gate before 10:10 (boarding deadline).
That seems quite tight, but there are quite a few variables:
- Is the flight on time/early/late? Recent history for the same flight, but arriving at 7 am, shows arrivals ranging from -15 minutes to +30 minutes. Not good.
- Do you get a jetway? I couldn't find the info.
- Are you seated towards the front of the place?
- Are you a quick walker, not encumbered with lots of luggage, kids, strollers, and whatnot?
- What passport do you have?
- How long will the queue be at immigration?
- How long will luggage take to be delivered?
I honestly wouldn't count on it. If you can change to a later flight, do it. If you cannot change it, book another flight. You may still try to catch the original one and "fall back" on the alternative at the last minute, but it'll probably less expensive if you buy it now rather than at the last minute. You'll also have the peace of mind to know that you have a seat available.
The big issue here is that flights from Athens to Skiathos are quite unfrequent. You may have to wait a full day at the airport, or wait until the next day.
One other important point to note: for some airlines, if you miss a flight, all subsequent flights on the same ticket will be automatically cancelled (you're supposed to use the flights "in sequence"). If that is the case for your ticket and you have a return flight, it may be better to change the original flight rather than miss it an use another flight. But it all depends on the costs of the different flights, the type of fare, and policies of the airline.