Singlespeed & Fixed Gear - SS cog from an old cassette

Just got my first SS (GT Peace 9r), and had my first ride yesterday thru the snow. The ride along the dirt snow in packed snow was fine, but when I hit a flat trail that was semi-packed from walkers and skiers, I had a heck of time. The gearing on the bike is 36x18. I think if I switch to a 22 cog it would help in the slow going of the snow trails. I have an old cassette in the parts box, just wondering if I can take the cassette apart and use a cog from there? Its a 9 speed cassette, so I figure the teeth of the cog will be narrower. Or should I just go buy a SS cog in 22 size. Given the position of the axle, I think I'll need to buy a new chain to handle the new cog, so I need to place an order anyway.

36x22 is low. try and stick to unpacked snow, semi packed it the worst!

Yeah, you can use an old cog from a cassette. You might have problems dropping the chain though. A cassette cog with its ramps, and short twisted teeth will magnify any problem in your drivetrain. However, it will work without issue if everything is setup correctly.

36x22 is pretty low. It might be great in the snow, but don't plan on riding on too much pavement!

I ride the same ratio on my snow bike, seems to get the job done

Thanx's for the help. The current 18 cog seems like it will be great come spring on dirt roads and trails, but pushing snow was tough.