The unfortunate thing about hair growth is that it tens to grow at about the same rate for most animals.
Hair goes in cycles, Anagen, Catagen, Telogen. These are roughly, Growth, Transition and Pause phases.
The first step to faster hair growth is to skip these other two. The first phase lasts 2-6 years. Catagen is short and lasts a couple weeks at best. The last can be up to 4 months. So if you got those out you get constant growth, since this last phase is usually when you shed hair. This could be a genetic quirk lets say.
Moving on to the next issue. Hair grows at about 15cm (~6inches) per year. There aren't really any weird quirks that change this. You could simply claim genetics or magic for faster growth, or you could use some kind of substance.
Genetics would mean that the elves have more blood flow to the follicles and more nutrient delivery. You could then balance this with the lack of any hair on their bodies, or maybe lack of finger nails or any other similar lack. Honestly you could just avoid all that tho because it doesn't take much in the grande scheme to feed hair. They could simply eat a tiny bit more. I hate hair so I'd go with the lack of any other hair on the body :D
Magic, is magic so no explanation needed.
As for substances, in the real world there are some hair growth stimulants. Minoxidile is one. This one works as a vasodilator, it opens up the veins. This helps but it also seems to trigger the hair cycles to shift. So hair sheds and starts to grow again. There however isn't really a faster hair growth compound out there.
That doesn't mean you can't have a substances that's made up that helps the elves grow their hair thicker, faster, stronger, and maybe even magically infused or capable of storing magic. This is a Fantasy story after all.