What are supplementary exercises one can do to increase sprinting speed? I've been working on my core and hip flexors so I can pull up my legs faster as well as squatting to push harder. It seems like the squats are not so helpful since they use the hamstring and require you to push from your heels, whereas in a sprint you're almost never on your heels. Basically, I've noticed that the weight on the squat bar has almost no correlation to my sprinting speed. Are there any replacements that people use to increase sprinting speed beyond those I've mentioned?
You could try plyometric weighted squats. Basically, jump squats, but with a weight on your back (or front if you prefer). I've found these to be excellent when trying to improve speed as you build a lot of explosive power, which is exactly what you need when sprinting.
What are plyometrics?
Plyometrics, also known as jump training or plyos, are exercises in which muscles exert maximum force in short intervals of time, with the goal of increasing power (speed-strength). (Source: https://en.wikipedia.org/wiki/Plyometrics)
Procedure: Go down in your regular squatting form, and come up about 1/3rd the way like you normally do. The rest of your ascent should be executed as fast as you can, followed by pushing off the ground with your toes (jumping), and then landing back gracefully. Keep in mind not to land back flat on your feet, but to start on the front of your foot and then land the rest of your foot back flat as you finish the landing (or you'll end up jerking your joints too much)
(Try not to be explosive from the very bottom, as that may lead to excessive strain)
Of course, start with a really light weight (when compared to your regular lifts). My regular back squat is ~350 lbs (about 160 kg) and I still don't go more than 100-110 lbs (~50 kgs) with the explosive ones.
But yes, they're excellent for building power and speed. Try them out and let us know how it goes.
Based on personal experience, the big lifts do help a lot. Squats and deadlifts followed by a plyometric superset (e.g. vertical jump, or broad jumps) - the key here being not training to failure, but training until you feel your form starting to deteriorate.
But the one thing that I noticed help me get faster: Sled pushes and pulls. The biggest reason for this would be because it strongly mimics your body position when running, and if you're intentional about it, you can practice your hip drive as well.