It's simple to get a cab at any hour from Santiago. Your hotel can call one the night before.
I will be on that 6:45 am flight from Santiago to Madrid in July, if all goes well, connecting to a flight to the US. But it's all on one ticket. It sounds like you already have your return flight ticketed from Madrid. If that's not the case, look at open jaw tickets -- several years ago someone on the forum suggested I do that, and I found I could add on the Santiago-Madrid segment for a trivial amount extra. That gives you protection, too, because if it's all on one ticket and the flight from Santiago is late and you miss the connection, the airlines will rebook you. If it's a separate ticket, they won't, you'll have to buy a new ticket home from Madrid. But the odds of that happening are pretty low, given that the flight is scheduled to arrive at 8 am, and your flight leaves at 11 am. One thing that could complicate it is if you have to change terminals, pick up and recheck bags, etc.
So, if you are flying on Iberia into Madrid, and Iberia or a One World partner out of Madrid, it will be easier to make the connection because you will be in T4.
Staying at an airport hotel may be a kind of depressing way to end a glorious month walking across Spain, and maybe you won't be a "big chicken" by then, since after all you will have walked all the way across Spain by yourself. Getting to the airport T4 from downtown
It's simple to get a cab at any hour from Santiago. Your hotel can call one the night before.
I will be on that 6:45 am flight from Santiago to Madrid in July, if all goes well, connecting to a flight to the US. But it's all on one ticket. It sounds like you already have your return flight ticketed from Madrid. If that's not the case, look at open jaw tickets -- several years ago someone on the forum suggested I do that, and I found I could add on the Santiago-Madrid segment for a trivial amount extra. That gives you protection, too, because if it's all on one ticket and the flight from Santiago is late and you miss the connection, the airlines will rebook you. If it's a separate ticket, they won't, you'll have to buy a new ticket home from Madrid. But the odds of that happening are pretty low, given that the flight is scheduled to arrive at 8 am, and your flight leaves at 11 am. One thing that could complicate it is if you have to change terminals, pick up and recheck bags, etc.
So, if you are flying on Iberia into Madrid, and Iberia or a One World partner out of Madrid, it will be easier to make the connection because you will be in T4.
Staying at an airport hotel may be a kind of depressing way to end a glorious month walking across Spain, and maybe you won't be a "big chicken" by then, since after all you will have walked all the way across Spain by yourself. Getting to the airport T4 from downtown Madrid (Puerta del Sol) is a real snap on the Cercanias train line and there are many reasonable places to stay near there. But there are lots of airport hotels, I'm sure people on the forum have some to recommend. Buen camino, Joodle.