If you are looking for English Teaching work one option would be to contact all of the people who offer English teaching jobs year round.
1). Call them to set up a face to face meeting with one of their representatives.
2). Email them your CV/Resume so they might have a copy handy for your meeting. (It does you no good to send them your CV/Resume before they are expecting it).
3). Meet with them in person to review your experience and availability.
4). Keep them up to date on your availability as it changes.
They will call you as classes come in and the people they usually count on can not accept them.
The more people you meet with the better your odds are of getting work, so just keep spreading the word that you are currently here in Madrid and available for work.
Also you can place free ads both on-line and in the published versions of the following publications, etc.
In Madrid www.in-madrid.com
Broadsheet www.spainalive.com
Segundamano www.segundamano.es
Aprende Mas www.aprendemas.com
Also, you should look at all of the classified sections of the above publications, etc. for job offers as there are a limited amount of English language resources and you will easily be able to see who is hiring or looking for people at any given time and sign up or remind them of your availability.
Rocco.

There are so many people offering English Teaching work in Madrid that it may cost you a bundle in phone calls just to reach them all and set up your face to face interviews.
Because of this high volume of people to contact, you will need to make a priority list based on what you might have heard about each one of them and considering your work status and if they hire non European Union citizens or not.
Then once you have decided who you will call it will do you no harm to just e-mail the rest of them to see if anything comes of it.

1. Everyone can teach private lessons arranged directly between you and the students. This would be classes through flyers, free ads in Segundamano, Aprendemas, In Madrid, etc.

2. If you are an EU citizen you can apply to work with all of them.

3. If you are non-EU / illegal you can still find work without having a visa and there are still people who will hire you.
Just make "Do you hire non-EU teachers?" your first question before wasting anymore time on the phone with the people you are calling.

Rocco.