Perhaps the fastest way to grow an Instagram account is to use already-existing networks of influencers in your space.
There are marketplaces where you can contract Instagram influencers by the hour or by the day to post promotional posts for you. Two of the biggest are Shoutcart and Instafluence.

These work very well if you know how much revenue you're going to bring in and are a little bit farther along, but they can be very expensive, especially in the beginning. Instead of using these networks specifically, we're going to reverse engineer that method and build out our own ad-hoc influencer network to drive down the costs.