That's probably the Hotel Dulcinea.
On your arrival you will probably be pretty wasted from many hours travel, so my advice would be to continue to Cordoba or Sevilla (whichever) when you arrive, have something to eat and then 'crash' for as many hours as it takes to get your body clocks adjusted.
Then you can work your way back to Madrid at a leisurely pace.
On Iberia, we once went Madrid-Sevilla, Sevilla-Barcelona, Barcelona-Vigo (Galicia) and Vigo-Madrid on a 4-segment 'visit Spain' pass. We weren't in a hurry - it was just the fact that the pass was so inexpensive. Spanair also has some pretty good deals on their Madrid-Lisbon service. So flying can get you a lot of places in a short time; but since you are already committed to staying in Madrid ...