For those that want a more old money looking neighborhood, you have your previously mentioned areas like Sedgwick(city's North Side):
https://www.sedgwickfarm.org/
https://www.sedgwickfarm.org/homes.php
https://www.google.com/maps/@43.0620...2!9m2!1b1!2i37
Strathmore(SW corner of city):
https://www.strathmoreneighborhood.org/
Strathmore
https://www.google.com/maps/@43.0227...6!9m2!1b1!2i37
Scottholm(city's East Side):
https://en.wikipedia.org/wiki/Scottholm,_Syracuse
https://westcottsyr.com/tours/scottholm/
https://www.google.com/maps/@43.0422...2!9m2!1b1!2i37
but there is also Berkeley Park(city's East Side near Syracuse University):
Berkeley Park
https://en.wikipedia.org/wiki/Berkel...toric_District
https://www.google.com/maps/@43.0315...2!9m2!1b1!2i37
some homes on East Genesee Street near Scottholm(East Side of city):
https://www.google.com/maps/@43.0445...2!9m2!1b1!2i37
Genesee Park Drive(East Side of city):
https://www.google.com/maps/@43.0443...2!9m2!1b1!2i37
parts of the Bradford Hills neighborhood(East Side of city):
https://www.google.com/maps/@43.0362...2!9m2!1b1!2i37
https://www.google.com/maps/@43.0372...2!9m2!1b1!2i37
https://www.google.com/maps/@43.0383...6!9m2!1b1!2i37
Shotwell Park/Hickok Ave. area of Eastwood(NE corner of city):
https://www.google.com/maps/@43.0622...2!9m2!1b1!2i37
https://www.google.com/maps/@43.0638...2!9m2!1b1!2i37
https://www.eastwoodneighbor.com/ (James Street serves as a Main street for the neighborhood, which has about 10% of the city's population. There is a small collection of businesses down the hill to the south on Burnet Avenue as well.)
A portion of the Park Avenue neighborhood(east of Geddes closer to Downtown on city's West Side):
https://www.google.com/maps/@43.0520...2!9m2!1b1!2i37
https://www.pinterest.com/pin/137570963587554305/
Besides Downtown, you have a park, Middle Ages Brewing, the Polish Home, an Aldi, Syracuse Academy of Science Charter(grades 7-12), Sacred Heart Church(historically Polish parish), Harrison Bakery, a couple of credit unions and perhaps even the pubs/few places to eat in Tipperary Hill(historically Irish neighborhood) that you can walk to.
Dewittshire(town of DeWitt/Jamesville-DeWitt SD just east of the city):
https://www.facebook.com/DeWittshireNY/
https://www.google.com/maps/place/De...!4d-76.0696493
https://www.google.com/maps/@43.0343...2!9m2!1b1!2i37
Can walk to this plaza, a Tops Supermarket, another plaza that includes a few Indian businesses or a synagogue on East Genesee Street:
https://www.google.com/maps/@43.0353...2!9m2!1b1!2i37
There is also Moses DeWitt Elementary in the neighborhood, which is a solid/good school with a decent degree of diversity(culturally and even economically):
https://www.jamesvilledewitt.org/mosesdewitt
https://data.nysed.gov/enrollment.ph...d=800000041065
The Orvilton area across East Genesee Street(town of DeWitt/J-D SD):
https://www.google.com/maps/@43.0368...6!9m2!1b1!2i37
I'll stop there for now, but what is could be appealing for those coming from NYC or other bigger cities is that most of these areas offer a decent to good degree of walkability and/or access to public transportation(bus).