Well, this story takes place after the beauty pageant mentioned in the first part, so it is indeed a bit unlikely that her hay fever lasts that long. But considering Asaki is about to graduate and how she talks about Tsune and their relationship in her inner monologue it actually sounds as if quite some time has passed. Maybe it's an entire year, so it's pollen season again.
Anyways, no need to get upset over such a small detail.
I liked the story (both parts, I mean) a lot. Tsune especially, she doesn't seem like an overly cliched tsundere. It seems she's very shy when it comes to communicating her feelings, but not automatically as shy when other things are concerned. When there's reason (warning another about a perv, having some small talk, explaning obviously odd behavior of others), she does talk, which makes her more realistic. And yes, I can connect with such a character on a very personal level, since I'm the same (though I do talk a lot more once I start, hehe).