In cases like this, I often find it helps to create a spreadsheet in Excel to make a timeline. You can record what records you have and under what name, and the date associated with each fact/document -- then you can see at a glance the window of opportunity for the name change. (This is my standard procedure for tracking women who have remarried, which is not unlike the scenario you describe.)
I found a record of naturalization for one of my husband's relatives, but did not know the date. However, the reference I found did give his current address. I made a list of all his known addresses in my file with dates, and was able to narrow down the date of his naturalization to a period of about three years -- still cumbersome to search, but far better than my starting range of 'sometime after he entered the US and before he died'.
So just keep logging bits of information from ALL your sources and see what you can find.
Are you looking for a whole family, or just one individual? Does your focus person have siblings? Don't neglect their records! When I was researching my husband's grandfather, I was able to find out a great deal of information about his family by looking at records about his older and younger siblings.
For instance, might any of the family members be in the WWI Draft Registration lists? If you don't have access to Ancestry, try looking on FamilySearch's new Record Search.
Especially if you have an Ancestry subscription, don't neglect Steve Morse' One Step Web Pages -- they greatly simplify looking for people on Ancestry and elsewhere when you don't know the surname, and looking at Steve's list of search pages might alert you to major collections you haven't tried yet.
http://stevemorse.org/
Hope this helps.
Jan