It is curious whether this was the original intent for the series, considering that Hawkeye and several other series productions were delayed by several months due to the ongoing pandemic. However, late-summer or early-fall could have easily been faked to take place around winter.
On the MarvelStudiosSpoilers subreddit, Reddit user /u/jkconhen626 posted several pictures from the set of Hawkeye filming outside their apartment. None of them stood out in particular, except for a man seen wearing a classic Santa Claus outfit, indicating that, at the very least, this scene takes place close to Christmas
📷 | Hailee Steinfeld and Jeremy Renner filming “Hawkeye” on December 03 in Washington Square Park, New York.
(Credit: jkcohen626 on Reddit) pic.twitter.com/6HqEXLJ0Lr
— Hailee Steinfeld News (@HSteinfeldNews) December 3, 2020
More photos were captured, posted by @BRMarvelNews on Twitter, showing Jeremy Renner, Hailee Steinfeld, and the dog playing Lucky walking in the park.
Steinfeld is no longer wearing any purple under her trench coat, and the jeans she’s wearing are torn and have the legs rolled up. It seems like they’re a tad too big for her and that the pair of jeans maybe belongs to Barton. It’s hard to tell when this scene exactly takes place in relation to the one in the subway since Clint seems to be wearing the same outfit he wore in that scene, but his backpack is missing.
— Marvel News (@BRMarvelNews) December 3, 2020
It could be that whatever reason they were being chased, it may have had to do with what was in the backpack. Clint maybe lost it to whoever was chasing him and Kate, or maybe he stashed it somewhere and had Kate change out of her purple-colored underclothes.