Can I use coconut milk without a refrigerator just for the day?

Look in your grocery store, probably either where breakfast cereal or canned and bottled juices are. There are juice box size containers of almond milk, coconut milk, rice milk and soy milk. Natural food stores like Whole Foods are an especially good place to find these. Each box holds about 8 -10 oz and you can sometimes get 3 and 6 packs. These containers are generally pasteurized and don't require refrigeration before consumption. Since they are fairly small, you can use one container and throw it out when done and use a second or third later on the second flight.

If you can't find these smaller containers, consider partially freezing one and completely freezing a second or third container for later consumption. They will be fine for your 6 hour trip.

And, depending upon how young he is, you might be able to find soy formulas that can be mixed with water. These also do not need refrigeration.

Just make sure any containers you get meet FAA regulations for liquids. Since it's baby food/formula, I think you can bring more than 3 oz of liquid inside. Getting the soy formula, you wouldn't have to worry about that.

use a travel cooler to store it cold u can get small fabric coolers for bottles with ice packs that don't take much room or taste the milk before feeding but my guess is it wont go bad in that amount of time

If you bought it on a shelf, it has an expiration date longer than one that is taken from a grocery refrigerator shelf.