There are a few ways to go with this. One is to simply upgrade the components on the existing crossover to higher quality parts but keeping the values the same, especially replacing any electrolytic caps with film types. That usually yields a subtle improvement without completely changing the speaker's tonal character.
The other method is a mod where there is something specifically wrong with the speaker you are attempting to fix. With this method you not only upgrade the components but you adjust the values to compensate for a shortcoming that you have identified. This may take a lot of work and experimentation to get right. If somebody has already did the analysis then it's a easy process as long as you have a basic understanding of soldering.
Next, is building a crossover from scratch. That takes the most work and usually is not necessary. As you have noted, buying an off the shelf crossover and slapping it in never works right. A crossover must be tuned specifically for the speaker it's used in. If you buy a PE crossover it's a crapshoot. Who knows what it will sound like. You may get lucky and it might be an improvement but odds are it won't.
Be careful about some of the supposed crossover upgrades. Sometimes they help some aspect but screw something else up.
BIC DV62si speaker mod for warm musical sound. PM me for info.