I've worked with two startups now, both with low support volumes, which grew many times upon acquisition. In both cases, these acquisitions brought upon an influx of new customers. It's simply bad business to scale your support staff based on MRR in the long term.

When the tipping point between growth and headcount arrives, a dev is usually assigned to figure out how to slow the incoming requests. There are many ways to do this, and some of the best companies have figured it out and mastered it very well at scale (see Google, Facebook, etc).

Some ideas:

1. Offer self-help/self-resolution documentation or tools (Gmail's self help is quite good!).

2. Categorize support requests, capture low hanging fruit, deploy fixes (sign up deficiencies, UI bugs, etc).

3. Set up a status page to proactively notify customers for outages/maintenances.

4. Increase customer communication when large functional or UI changes disrupt users.

5. Automation/AI/Bots (Not easy, and hardly ever done right)

6. Offer chat to increase interactions served by a single CS agent.

An interesting method, that is often heavily debated... Only offer support to paying customers. I think businesses can only get away with this, if they offer plenty of self-help documentation/tools, and your core product is relatively rock solid. Operating on a freemium model always brings about interesting challenges.

Edit: Formatting.