Pro active supplemental learning should be encouraged and only tempered by proper instruction when it applies to your technique or anything related to the way you control your body, arms, wrists, and fingers.
Managing the precious surplus time you have and aiming it at practicing scales, arpeggios, and chords, learn new ones, improvise with what you have already learned, and so on can only help to master your craft.
Also consider performing with others in an ensemble as a way to practice listening, communicating, and collaborating, as it will facilitate a higher level of mastering your instrument.
While singing in a choir may not help you make better octave reaches with your fingers, it will improve your overall ability to read, listen, and make music. Remember music is a highly collaborative artform. Even a soloist will tell you that they are collaborating with the composer to interpret the music to the audience.
However, when if comes to such areas that need instruction e.g. the correct fingering for a passage in a new piece or the proper way to execute an accelerando in the context of the literature, it may be best to wait for such instructions as you may have to 're-learn' what you taught yourself. Although, youtube may have some pretty good examples of the 'correct' way to play so and so's sonata, fugue, nocturne, etude, or fantasy, this may be the exception. Best to ask your teacher for any such links.
For now, I suggest you work with your piano teacher to set reasonable boundaries on what would benefit you most with self learning vs what you should wait for with regard to instruction.
At the professional level many pianists will hire an expert teacher best suited for the repertoire they are preparing. Although you can learn to master the piano, the learning will never end, there is always something new to learn, something inspiring to aim for.
If you get to the point where you think you know it all, and can play it all, it may be a good time to start to think about being a painter.
UPDATE: I just want to be clear that I have nothing against painters, actually I very fond of their centuries of great works. It may be more clear if I had written, if you think you know it all... try something new like painting as it is wide open, will humble you, and put you at the beginning again...a fresh look as in the Zen of a beginner, may actually help you move to a higher level on the piano as well.
