When it comes to performance, there comes a point where throwing hardware at the problem will no longer be a viable solution -- it'll only increase your costs and provide short-term relief.
Since you've tagged Apache in your OP, I'm assuming you're using a LAMP Stack? If so, the best way to boost performance, or attempt to with the hardware you have, is to work on tuning your stack.
Take a look at tweaking the configuration for Apache, PHP, and MySQL. The stock configuration is really only meant to get you up and running. You can run production sites with it, but they may not perform as well as they could or would once you've tuned things a bit.
Also, look in to caching -- Memcached or Redis. WordPress has a number of plugins that support both and Laravel provides out-of-box support via drivers. You can run individual instances for caching which won't effect your primary server, so for example, you could deploy a 1GB Droplet, install either Memcached or Redis, and connect to it remotely.
You could also look in to separating things out - i.e. one server for Apache/PHP, one for MySQL, and one or two for Caching (depending on need).
There's a number of things we could do and there's always plenty of options. Scaling vertically is not always the best resolution, sometimes you need to scale horizontally.