Assuming it conforms to the classic Dynacomp/Ross design, there will be a 27k fixed resistor on pin 5 of the CA3080 chip. This sets the upper limit on how much current can be fed to the Iabc pin. The Compression pot adds up to 500k of series resistance on top of this. To reduce how "sensitive" the 3080 is to this current, you can do a few things. You can increase the 27k resistor, or you can increase the 500k pot; both will result in less control current hitting the Iabc pin. It may be that moving up to the next standard pot value (1M) is a bit too much so consider starting out by replacing the 27k resistor with a 39k or 47k unit and seeing if that makes a difference in how much subtlety you can achieve.
The "attack" control is nothing of the sort, even though many commercial designs have used one and labelled it attack. Technically, it is a "recovery" control, and determines how quickly maximum gain is restored following a sudden transient. With a very quick recovery time, maximum gain is available right away and lets you hear the initial pick attack of every plucked note. With the slower stock recovery time of the Dynacomp/Ross design, picking the first note in a run hard forces the gain to be reduced, and since it is still taking a bit of time to be restored after that, any notes you quickly pick subsequent to that will be at less than maximum gain, so they sound like you have "lost attack".
The component to alter is the 150k resistor tied to +9v (the only one on the board to do so), which provides current to charge up a 10uf cap. As it gets smaller, the 10uf cap it is connected to charges up faster, restoring maximum gain sooner. On all those commercial designs that use it, the range covered is essentially 10k (min value) to 160k (10k + 150k custom pot). You might consider going higher than 160k, but quite frankly the breathing may be objectionable if you do so.
There are 2 ways to cover that resistance range. One is to score a custom pot, or get yourself a 250k pot and stick a parallel fixed resistor in to set the max resistance. A second way is to use a toggle switch to select presets. My buddy Tim who makes Retro-Sonic pedals ( http://www.musictoyz.com/guitar/pedals/retro.php ) pursued the latter at my urging, but seems to have switched to a pot more recently because customers seem to want a continuously variable control, even when it does them no more good than a 3-way toggle.
To get slow, medium, and fast recovery, you can do the following:
1) Get a 3-position on-off-on SPDT toggle (or DPDT but only use one set of contacts)
2) Wire up a 10k and 150k fixed resistor in series and use that to replace the 150k unit already on the board.
3) Run a wire from either end of that 150k resistor to the centre lug on the toggle.
4) Solder a 39k resistor between the two outside lugs of the toggle.
5) Run a wire from one of those outside lugs to the other side of the 150k resistor.
What you now have is a switch that will either place nothing in parallel with the 150k (slowest recovery), a 39k resistor in parallel with 150k (=31k in parallel =41k total with 10k) for medium recovery rate, or a wire link in parallel with 150k for fastest recovery time (just the 10k on the board).
Small variations in recovery time are VERY hard to notice, so most players will only use the extremes anyways. This 3-way switch provides a middle. Note that the values are selected to give an approximate 4:1 change in each position (10k vs 41k vs 160k), so that it is psychologically the "middle".