How can I catch better fish in Hades?

There's a fishing-related boon but it only increases the number of fishing points for the run, not the chance to get a rare one.

There are two parts of catching a good fish: timing and luck. There are two possible types of catch, and only the second one has a chance to give you a rare fish.

1. When you're kinda slow and you get a "Catch!", you can only get one of the first two tiers of fish for this biome: "common" and "uncommon" (I don't think there's an official name for the rarity, e.g. for Tartarus common is Hellfish and uncommon is Knucklehead)
2. When you're fast enough to get a "Perfect Catch!", you'll get either an "uncommon" or a "very rare" fish, which for Tartarus is the Scyllascion. I'm not sure of the exact percentages, but it's close to 50/50 - the name "very rare" is kind of misleading, I've had runs with 3 fishes that were all "very rare".

Maybe you've been getting perfect catches and were just out of luck, but in case you're only getting basic ones (as was my case for a while) and to help anyone who might read this later (mostlty relevant for keyboard users, from what I know it's much easier with a controller due to the vibration), here are two tips that have helped me getting many rare fishes:

1. There can only be a maximum of 3 "fake bites", so when the bait goes down for a fourth time it will always be a good catch. Knowing that should help you be faster and get a perfect catch at least 25% of the time (assuming the "real bite" is distributed randomly)
2. A bit more tricky to use, but I've noticed that the more you wait for a bite, the more likely it is to be fake. So when it bites early you should click the button, but after a few seconds (I haven't properly timed it, you learn it by experience) you should relax and then start again with the next one.