You're in the right place to make the most of networking from within an airline rather than outside, so make the most of it if you can.
Any chance you have to get free sim rides / sit in the back on a sim check with flight crew / get exposure to flight ops / jumpseat for landing would all be a benefit to you.
In this day and age, you might be able to arrange a sabbatical to do flying training in one go (though at a later date), and retain your present job - curry favour with them in going above and beyond the call of duty, and you might even be the first one they call when they hire flight crew once again...