When I'm trying to figure out a color's season, and I don't have swatchbooks in front of me, I try going through the three dimensions of color one at a time.
I ask myself "Is this bright or soft? Is it light or dark? Is it warm or cool?"
Sometimes I'm stuck on one of those questions, but answering the other two makes things start to become clear.
This is all harder to do with lights and neutrals, but in those cases I try imagining alternate versions:
"Could this be dirtier? (More greyed?) What would that look like?"
"Could this be purer? (Less greyed?) What would that look like?"
"Could it have more yellow? What would a more yellowed version of this look like?"
"Could it have more blue? What would a more blued version look like?"
Warm and cool -- yellow/orange-tinted or blue/pink-tinted, to be simplistic about it -- can be tough to decide until you've memorized a warm and cool version of each hue. Once you have those mental images, it becomes easier to determine temperature because you can compare a sample to those mental images.
For example, for brown I have memorized mental images of caramel, a warm brown, and cocoa, a cool brown. When I see a brown in a store, I can ask, "Is this closer to caramel or cocoa?"
For blue, I have memorized mental images of aqua (a yellowed blue) and periwinkle (a purpled blue).
I can analyze whites pretty easily without a swatchbook if I can first identify what the white is tinted with.
So, for example, if I figure out a white is blue-tinted, I can ask myself, "Is that drop of blue more aqua-ish or more periwinkle-ish?"
It's also a good idea to compare colors in a store to other nearby colors.
See the red in front of you: is it the brightest red in the store? Or is it "dirtier"-looking than many other reds? Does it seem to have more orange in it than other reds, or does it seem to have more violet?
Commercial products, like Coke cans, are great reference points for this kind of comparison because they are often purely saturated. If you're looking at a red that's as bright as a can of Coke, chanes are it's a Bright season red.