First and most obviously, mine in pristine metallic rings. I'm sure you know, I'm just trying to be thorough. Just pick a random spot, no hot spots.
Mine the small yields. They add up. I mine anything over 10% for osmium. You're only after 40 tons. If you bypass 40 asteroids with osmium to mine 3 high-yield ones, you'll have 12 tons of Osmium when you could have been done.
Use your pulse wave analyzer, check the yellow asteroids. Every surface deposit or fragment from a subsurface deposit is close to a ton by itself.
Be patient. In the time it took me to pick up 62 tons of platinum to participate in the recent community goal, I picked up only 19 tons of osmium (and 33 tons of palladium, for reference--that was in a platinum hot spot, though; the ratios might have been better elsewhere). I think I spent two hours, but it might have been 3. I didn't keep track.