I will often (carefully!) stop a fan with my finger to see how much effect it has on the noise signature. It doesn't have to be the loudest to be the largest contributor. Pitch and tone matter. If you can put your HDD to sleep, you can easily judge its contribution too.
My technique for quieting old PCs without turning them into new PCs in an old box:
1) PSUs. Old power supplies had kinda noisy fans that ran at full speed with metal grills that were not in the least aerodynamic. I don't even bother to test old PSUs anymore when I buy a new computer or new-to-me AT supply, I just refurbish them with new caps and a new fan as a matter of course. Then test them. Note that the PSU is an active component in the airflow of an old PC. If you're going to replace the fan with a lower-speed (ergo, quieter) fan, you will probably increase the temperature of the case. Deal with this by adding additional low-speed intake and/or exhaust fans, or if the case temp is still tolerable, let it be. Note that the speed and blade configuration of a fan both contribute to its noise profile, and not all "quiet" fans respond well to being used in a PSU with typical (for the era) circular cutouts in the rear panel for the grill. The wire grills we use now are much better in this regard. Point is, you might need to try a couple of fans to find the right one, as a free-air test won't be a true indicator of its volume once installed in a PSU. And/or, cut out the entire stamped grill and install a wire grill instead.
2) CPU fans. Pentium IIIs are in a crossover period. Fans got much quieter in the P4 / Athlon XP era. They had to, because those CPUs put out a lot more heat. Old CPU fans revved up high enough to deal with that heat sounded like they belonged in a server room. They also weren't typically thermally-controlled. So, if you're still using the stock fan on a PIII, chances are replacing it with a new one will cut down the noise quite a bit. Depending on the clock speed and airflow situation, you might be able to get away with a low-speed fan on a nice copper heatsink. The down side is it's most likely going to be a 60mm fan (which is not quiet as popular, and so you have fewer options) or a 40mm fan (which has to spin faster to make up for its lack of surface area). Worst case is 50mm, which is an uncommon size -- you may get stuck with whatever you can find that fits.
3) Case fans. Get the largest, slowest ones you can find and that will fit. If you have to overcome heat, add more. More slower fans beats fewer noisy fans. Clean up the cabling as best you can to reduce impedance to air flow. If the case has stamped grills, consider modding them to wire grills. If there's copious front bezel plastic around the intake, you will be fighting wind noise. You might be able to line the back side of the front panel with something to damp the sound. Take it case-by-case, no pun intended.
4) Hard drive. Once we got to 20GB, drives started getting considerably more quiet. Fluid bearings showed up around this time, which made a dramatic difference. Some were better than others, though. (I brought home a brand new Maxtor that was twice as loud as equivalent Seagates at the time.) By the 40-60GB era, any quality drive was fairly quiet. By 80GB, better still. I'm unwilling to choose drives outside of period-correct hardware, so you won't find any 160GB drives in my PIII, nor SSDs. But if you're not as picky, that'll all but eliminate hard drive noise. Decoupling it from the case, through shock mounts or insulating hardware, will also help.
5) Video card, chipset, etc. You already mentioned your video card is passively cooled. Great! PIII chipsets won't usually have a fan either, so you're probably already done then.