Thank you Col, Roger and Clare for your advice, help and encouragement. I'm sad to report that I still haven't made much more progress. Although I have many photos of Winifred Phillips at the end of her life...and have some very, very fuzzy anecdotes from her grandchildren....I still have no additional good information to work with. I don't know her parents, where in Ireland she was from (only that she had a thick brogue), or exactly when she came to the US. I do now know that she married in the US in 1915 (to a fellow Irishman, whom she is said to have met here in the US).
I tried putting the County Cork rumor aside and working the twin sister angle for all of Ireland. I found one family on the 1901 census with twin daughters, with one named Winifred (even trying lots of spelling variations and possible nicknames). There is a Winnifred Phillips, aged 13 in Drumacoo, Edmondstown, Roscommon, Ireland. She is about the correct age, has a twin sister....but her name is Honoria. Family rumor says our Winifred's twin was Anna. I have no idea if Anna was a commor nickname of Honoria...sounds like it could be...but what do I know.
Since I have learned Winifred's date of marriage....I next plan to try to get her marriage certificate from Connecticut in the US and hope for the best from there. Perhaps there will be some good detailed information?! Onward and upward. ;)
Thank you all so much again. This is by far the best research website I have found...everyone I have come across is so kind, patient and supportive. I truly appreciate it. You have encouraged me not to give up.