Look through the entire City Directory, and not just the alphabetical or numerical sections of the directory. In the New England directories published by the Price & Lee Co., there's often a section with City Information. Social organizations are listed in the back with their officers and meeting times. These will not list resident aliens per se, but if the resident aliens gathered in their own organizations, looking for people who are mentioned as members of the organization in newspaper searches can yield a great deal of information; some of it might fill in the gaps left by the loss of the 1890 Census. The indexing can be very strange because the names are not presented in the same fashion as the rest of the directory, so it's better to browse the pages directly.
See this list of fraternal organizations on Wikipedia for examples. There is a Canada link on that page; if people were members in Canada, they might join the local group once they reached Chicago, or start one if one didn't exist already. Fraternal brothers often show up as bearers and can be discovered by newspaper accounts of funerals. Work backwards from later records and you might be able to reach the 1890s that way. Church newsletters may also mention the movement of members, as well as newspapers.
Newspapers sometimes have lists of names, including registered voters. I've seen lists of people who had letters waiting for them at the Post Office, property transfers, court calendars, and probate-related advertisements that could be clues to other records. and other residence information.
These pages on Olive Tree Genealogy may not be directly useful, but in case they are of interest: Lists Of Stockholders Canada & USA 1890 and Canadian Passport Records. This one is the wrong period, but perhaps you could find similar records which are from the 1890s? Names of Emigrants from the 1845-1847 Records of James Allison, Emigrant Agent at Montreal
I've also found city registers on Google Books which list official city business. They might give statistics on the number of resident aliens, and there might be clues to actual lists, e.g. the name of the department in the city which produced the statistics.