While discussing relationship-based honourifics, I went looking to see how the word for older sister, 姐, might have been pronounced in Middle Chinese. I checked the Baxter-Sagart reconstruction document (revised 2014), but they didn’t include this particular character, and the only one with the same modern pronunciation of jie3 was 解, reconstructed as “keaX” (in Baxter’s notation, the “X” is a tone marker, and is not pronounced*).
I was a little dubious, because only one character isn’t a good sample, especially when the orthography was nothing like the character I wanted.
Then, I noticed that most other characters with the element 且 are modernly pronounced with a z- (sounds like dsss) or c- (sounds like tsss) initial, found some of these characters in the reconstruction document, and confirmed that they were that way in period too. I wondered whether the initial might have shifted after the period when Middle Chinese was spoken.
First, I went looking for a Hakka pronunciation of the character, as their colloquial pronunciations often remain much more reflective of older pronunciation than modern Mandarin. Omniglot tells me that it is pronounced zia3. Then, I looked toward the languages that would have also adopted the character early on, Korean and Japanese.
Wiktionary told me that the hangeul writing for the hanja character 姐 is 저, romanised jeo. Going to the Cambridge English-Korean Dictionary, I confirmed that the word for sister is 자매. Not being familiar with Korean pronunciation, I went to a few websites that provided audio recordings, like Google Translate, and Forvo, the latter of which had two separate recordings from Korean contributors. All of the recordings gave a pronunciation that sounds like “zah”.
It was a big easier with Japanese. Wiktionary gave an on-reading of so, which several other online vocabulary sites confirmed.
From this, I am coming to the tentative conclusion that the character 姐 may have been more likely to have a initial of ts-, probably giving it a pronunciation more like tseoh than jie. Interestingly, this would make it sound more like the modern word 嫂 sao3, which, rather than “older sister”, means “older-sister-in-law”.
This is all conjecture, and I know there are other reconstructions out there, but none that I am able to access at the moment. If you have any thoughts or information on this, I invite you to share.
* Edit: our friend Þórfinnr (down in the comments) has clarified that the Baxter-Sagart reconstruction is meant to be a rime reconstruction, not a phonetic one.