If you want a pedal that does this automatically there are a number of options these days with different amounts of tweakability of the note envelope to allow you a level of speed and consistency that a manual volume swell would make really difficult. The general effect is just an envelope generator that triggers with a slow attack on the AR envelope. Names it can go by are “Slow Gear”, “Auto Swell”, “Attack Delay”, and a number of others (there doesn’t seem to be a consensus on this name convention!).
Boss came out with an effect called the Slow Gear that was one of the first popular versions of this effect but with limited tweakability (it’s been cloned and modified countless times and you can find models of it in lots of digital multi-FX like Line 6 Helix as well). EHX has the Attack Decay pedal, Malekko the Sneak Attack (both of these offer a lot of tweakability to get synthy textures), TC Crescendo (and countless other cheap basic Chinese swell and auto volume designs like it). There are also a number of delay and reverb pedals that have swell effects added internally- one of my favorites is on my Meris Mercury 7 Reverb, which has a dedicated Swell footswitch and control over the attack speed. And as you noted, lots of synth-type effects have “Attack Delay” engines like this built-in as well (I used the EHX Bass Micro Synth for this purpose for years).