where can I advertise as an English language instructor?

The European Commission staff (and the staff of other EU institutions) get language courses offered by their employer, free of charge and during working hours, so there would only be a limited interested to private lessons. This is especially true for English; as one of the three working languages (the others are French and German) there are always many courses of various levels scheduled and all those who want are allowed to study; this is not the case for other languages, for which the institution can (and often will) refuse the request for training.

If anyone is also looking for teaching jobs at the European Commission, they could apply to CLL. This is a private language school, affiliated with the UCL University, which has won a contract to send teachers to teach at the European Institutions. Note that they stipulate quite a few conditions, including 2,400 hours' teaching experience. Nevertheless, they are asking for practically all official European languages as well as outside languages too.
From my experience, it is possible to work through CLL whether your status is independent or SMArt member.

https://www.cll.be/cll-6/jobs/become-a-trainer-at-cll.htm?lng=en#other-o...