It's not a 'y'; that's the top of the number 6 (the number of children).

I tried the Allgemeines Verzeichnis but didn't come up with anything useful, and I don't think further staring at that handwriting will lead anywhere.

If those census entries really are all her, then I'm not sure we can trust anything that the family said about her origins: her ages make it look like censuses occurred every seven years, and she lost five years between her husband's naturalization petition (which says she was born in 1883) and her death certificate (which says 1888). The censuses also do interesting things with geography and language:

1910: age 28, married c. 1906, birthplaces "Aust Lith", immigration 1902, language Lithuanian
1920: age 35, immigration 1904, birthplaces Austria/Austria/Russia, language Yiddish
1930: age 42, age at marriage 17 (= married c. 1905), birthplaces Rumania/Rumania/Russia, language Polish, immigration 1904
1940: age 57, birthplace Austria

That's a lot of different mother tongues for one person (and yet she didn't speak English), and I don't know where in Austria-Hungary she could possibly have acquired any Lithuanian. (A-H was big, but not that big.)

1930's Rumania implies Bukovina, but the switch back to Austria in 1940 casts some doubt on that, as do the languages.

Half of the censuses agree on an immigration year of 1904, so that's the best I can come up with: try to find an arrival for her. Yes, I know, with a maiden name of Miller, that's easier said than done.