In the new Ancestry, I completely ignore the first page that comes up in the search, their new "montage" of pseudo relevant facts derived from family trees and other confused data. I go to Categaories in blue on the left and pick one, usually Census and Voter Lists if I'm just beginning. In the old Ancestry those +/- parameters actually limited a search rather than expanded it. I never used them. Re some of the suggestions above, I've been told by National Archives personnel, to search in the specific database of information rather than the general search for specific information such as Civil War Pensions, Ships Passenger Lists, Missouri marriages, etc. I can verify from personal experience that not all info in those databases makes it to the general search, but the specific search of the particular database will often turn it up. No idea why, probably coordination between the two search functions is not complete.
Re 1960 naturalizations, I'd be very surprised if those were in either familysearch.org or ancestry.com. They are public record in the Federal Courthouse in which the ancestor was naturalized. You may call or correspond with the office of the Clerk of that court to have copies of those papers sent to you for a reasonable fee. Since the recent records are pretty readily publicly available, they are not a priority for inclusion in familysearch or Ancestry.
It is helpful to read the descriptions of the databases in Ancestry and BTW any searchable database you're using to see what is included and what is excluded, especially if you're looking for County records and newspaper items. These databases bill themselves and containing much more than they do contain. If newspapers.com doesn't contain the Wichita Eagle for the year in question, you aren't going to find anything from that paper in that database.
Remember that your ancestor's name may be misspelled in the 1880 census. I was looking for a Ryan in the 1870 census just yesterday, and it did not come up under the general search. I searched the 1870 Census specifically narrowing the category to that census in particular and it did come up as Rynn. Why it didn't also come up as Rynn in the general search, I don't know.
The advice above from Allie 1st Lt. to narrow down to one census is good advice, as is the similar advice from others above. You might also get creative and try alternate spellings of your own devising -- we can't expect any search engine to come up with every variation.
Remember, these alternate spellings have nothing to do with your ancestor's level of education, they are the product of several factors: 1) the census takers asked orally for the information and then recorded what they heard, often not inquiring if the informer knew a correct spelling -- they could have mis-heard or used an alternate spelling of the name not used by the family -- in my family Burris for Burrows for example; 2) the census taker was not speaking to a person in the household who really knew the information but who approximated it, guessed or recalled incorrectly; 3) the census taker's notes were then transcribed again to make the census pages we now see, a process which results in the usual human errors in transcription from one person's handwriting into another's. Some census takers and transcribers are smarter than others. 4) then Ancestry or whoever indexes these databases (in China?) types them into something and we switch from handwriting to typing. Again the usual human typos and errors interject themselves into the process. As frequent users of Ancestry will all confirm, often the typed Ancestry summary of the original handwritten document contains misreadings of the original, and often it is the names that are misread. Some skeptics says this is due to the use of foreign labor to do these transcriptions and indexing -- the workers may know English but are not familiar with commonly heard US names. I've done work for myself and others for 10 years and I believe I have yet to see a person whose name was consistently spelled throughout all of their personal records.
Other reasons for alternate spellings in older records have to do with the low literacy rate and with lack of any requirement of conformity in spellings pre early 19th Century. Historians today tear their hair when they see an 18th C letter in which the family's last name is spelled three different ways by an actual family member in one document. Cluff, Clough, Cloph, etc. In my family, one branch spelled it Donal and his brother's branch spelled it Donnell. Same name, sounds the same, different spelling conventions adopted by different individuals, sometimes seemlingly on a whim.
Finally, if you want to find something DO NOT fill in all the specifics you know or use the search forms which Ancestry automatically fills in with everything on your family tree, just wipe that out and fill in at most name, approximate year of birth, location they lived without a timeframe and, if known, where they were born, then click "more options" and fill in sex, race and narrow the search to United States if it says All Collections, unless you are specifically looking for European or Canadian documents. The dramatically decreases the volume of irrelevant results returned.
When you fill in a lot of specifics, as Ancestry encourages you to do, probably for their own reasons, all of your specifics can be accurate, but too much detail prevents you from getting the matches you need which may not mirror your request. I usually just fill in what I mentioned above, and then narrow categories on the left side of the page as I search for different records.
Bringing the