You're obviously well versed in the book. Whether or not this is her first or second assignment, the newness of the menial chores is probably more for people like me who are only passingly familiar with the book.
Yes, I think the "this is new" vibe is for the audience's benefit, since it shouldn't be new to Offred.
This part actually makes more sense in the 1990 movie, since it's clear in that version that the Waterfords (they're never named in the movie, but Offred's name is given as Kate - which I prefer instead of "June") are her first placement, but of course there was another Offred before her. The movie is much more linear than either the novel or the TV series - the major scenes are there, but the ending is unlike the book (it annoyed me when a friend, who saw the movie before I did, blurted out the ending).
I'm honestly trying not to mention major spoilers here, since there's something coming up which the TV series hasn't gotten to yet. I am surprised that they decided to un-kill Luke, since he remains dead in the novel and the movie. It's nice to hear because I like the actor who plays him. It's going to be another 8 days before I can see that episode, though - Canada is a week and a half behind the U.S.
I was first exposed to Atwood's writing in one of my college English courses in the early '80s (her essays - she's an excellent essayist). Fast forward a couple of years, and then the Handmaid's Tale novel came out. I read it, and was utterly croggled. It's one of those near-future dystopian novels that could actually happen if the conditions were a little bit different.
I'd also been reading a lot of F.M. Busby's novels around that time, and one element in those was that the U.S. government could no longer afford its own system, so multinational conglomerates began bidding for the right to govern, and the citizens would vote not for candidates like now, but for which conglomerate would run things. One of those conglomerates just decided to take over everything - first North America and then most of the planet. It instituted Total Welfare - debtor's prison on a planetary scale, and which included tossing children into it and political prisoners.
So I was into dystopian stuff in the '80s, and one thing that I liked about the Handmaid's Tale was that the ending was ambiguous. And then a few years later the movie came out. I saw it twice in the theatre, and by that time (1990) I'd already read the book two or three times. I've read it maybe another half-dozen times since then.
Sometimes I'm just in the mood for a depressing dystopian story, and this is one of my go-tos for that - and it's a damn good reminder to keep fundamentalist religious nuts out of the government.
There are some good fanfiction stories posted on fanfiction.net (in the books section). I'm working on my own - a crossover with Sliders, of all things. Since I started it years ago, of course it's based on the book and movie and has none of the elements of the TV series (might change my mind on that, since it would be easier to write Rembrandt's part of the story; in the novel and movie all the non-white people are either sent to the colonies, deported, or killed).