A few guesses:
I will assume for the moment you have the Agil RITE (not the miniRITE) and you are able to flip into the t-coil mode (you said a little tune plays).
You didn’t describe what kind of a phone you put the magnet on. Is it a cell phone?
If it is a cell phone then a few issues can arise.
Cell phones that are useful with hearing instruments need to have certain minimum characteristics.
They need to limit the strength of the cell phone’s transmitted RF field in the vicinity of the hearing instrument.
They need to produce a sufficiently strong audio magnetic field that your t-coil can pick up. - (This has nothing to do with the magnet.)
They need to be sufficiently resistant to the interference the hearing instrument generates.
(Cell Phone manufacturers rate their phones for these properties. M4/T4 is the best rating. See http://aboutus.vzw.com/accessibility/digitalPhones.html)
If your phone it is not a cell phone but is a cordless phone, some of these issues may still apply (in this case Item 1 is not quite as likely to be an issue).
If your phone is a wired “land-line” phone then just Item 2 would be a factor.