Context:
This book was fiction and read from around 2014-2016 (paperback), but it seemed a bit worn, so the publication date is most likely before the 2010s. It was read in elementary school, but seemed like a higher level book since this was 4-5th grade and about to go into middle school (unsure if scholastic, I recall randomly finding it on a shelf, teacher might've brought it from home or someone donated it). About 200 pages (novel-like), it was small too and well worn, so it was used quite often and the pages were a bit tan. In English.
Can't recall the cover properly, so with a huge grain of salt, it had a lot of white on the front and back, with a figure walking into the storm (?), and some dark colors along the spine and edges.
Unsure of genre beyond just fiction since there's so few details to go off of, but I believe it's some fiction form of historical, literary, or contemporary. Maybe YA if it can count.
Details of the book itself:
Not too many I can recall, but the main thing that stood out to me was the main character who was male (unsure of character age, most likely young adult or below) going out into an active winter storm/blizzard for something that he thought was important enough to get severe frostbite. The thing that stood out to me the most was how in-depth the frostbite description was, detailing his hands changing color, how they felt, and how much time he had to get inside before he would get irreversible damage. He at some point collapsed or fell down, leading to his determination to keep going despite the frostbite worsening. Cannot recall for certain what the guy looked like, I'd assume white with an American name since I believe this book was set in America.
These details are shakier, but I believe the male MC went into the storm for either cigarettes or medicine. I also think the character was struggling in some way, either with his own personal struggles or with his family life, making him not care about the storm and risk his life just for a walk or to get something. No one else (reasonably) was outside during this scene, just the buildings and cars.
The setting itself (from that scene) was a long stretch of road that had plenty of shops and street posts along it. It was dark outside when the MC went on his errand/walk and got frostbite. I recall he was determined to keep going no matter what, even if it meant his frostbite worsening. Shaky detail, but I recall his hand was intact in the end, only severe frostbite symptoms that had to be treated and it was eventually normal again. I think it was set in the 90s/2000s, maybe 80s but I think it's highly unlikely.
These details were maybe only a couple pages of the entire book, and I struggle to recall any other major details of the book. I understand this may be impossible to find, and I thank anyone for trying :)
If I recall any more details, I'll add them with an edit below. Thank you!