Since you need clues from other record types in order to find the passenger lists you're seeking, some of the information here may also help with answering other questions.
Start a Research Journal
Write down what searches you've done, what wildcards you used, what surname variants you looked for, and other information you learn about as you try the suggestions in the answers and in research guides. Use whatever works for you, whether it is a free-form document or a structured report. I use Scrivener for its integrated outliner, index cards, document editor, and ability to include Scapple drawings; you may be more comfortable with writing a document in a word processor, using Evernote, or keeping a paper notebook. Use whatever works for you, but get into the habit of writing things down as you go.
Your goal is to leave a record of what you've done for your future self. Don't be afraid to write out explicitly things which are embarrassing obvious to your present self. Pay close attention to where you have "negative findings" because writing out why you might not have found something can help you later when you try a new search.
Review what you have
Start by following the advice in your related question. Build a timeline of 'known' events from all the sources you've found so far. Make a note of each event or fact and a note about which source the information came from.
It may also be helpful to make use of software like Evidentia or to create other 'helper' spreadsheets like the Crista Cowan's Genealogy Source Checklists video. Reviewing all previous research/discoveries can turn up clues in your sources that you might have missed before. You want to narrow the time period for the family leaving Ireland and arriving in the USA as much as you can, using any available sources.
Put what you know in context
Add to your timeline known major historical events that would have affected your family's life. An arrival around 1870 is well after the period of the Great Famine and pre-dates famine of 1879 whose worst effects were in the West. Understanding the "push" and "pull" factors that might have prompted the emigration to the USA might give you clues about the timing.
Study the record groups you are using
For any historical records we use, we can get more out of them if we understand the agency who created them, the purpose for which the information was collected, and the coverage (or lack of coverage) for the time period we are studying. For any database or website you use, read all available FAQs and research guides. FamilySearch has a companion article in the Research Wiki for all of their records; Ancestry has an "About the Database" section. NARA often has a Descriptive Pamphlet (DP) for a microfilm (found at the beginning of a roll, or available for download from their microfilm catalog) and sometimes there are Research Reports (RR) that can help you understand the records. Knowing how the records are arranged, how they were preserved, knowing how many times something might have been copied -- all of these things can help us find elusive records. Pay close attention to the start and end dates of any record group -- you can't find your people in a database if the event takes place before or after the time period covered in that set of records. If your people did arrive in 1870, they won't be in NARA's AAD Irish Famine passenger lists because that database covers the period 1/12/1846 - 12/31/1851.
Can you find other people in the community in Wisconsin who also came from the same part of Ireland? People rarely move alone -- they often follow other family members, or friends, associates, and neighbors (sometimes referred to as the "FAN Club"). Learning the path that other families took to reach Wisconsin can give you clues. Don't forget printed sources such as published travel accounts, or manuscript collections where you might find diaries and other letters. Look for Historical Societies and Genealogical Societies that cover your areas of interest.
Other immigrant communities
Sometimes you can get ideas about what information is out there to find by looking at the work of people studying other ethnic groups. If you read blogs about other immigrants to the US, what were the authors able to find? Did they use historical newspapers, immigrants' travel guides, historical maps, diaries? Use those articles to widen the type of records you are looking at.
Search "the same" database wherever you can access it, and vary your searches
Search engines differ. You might be able to find your people on one website but not another because of differences in indexing. Don't just search the NYC passenger lists on the Ellis Island Foundation's database and call it a day. If you don't have an Ancestry sub, see if your public library or genealogical society has Ancestry Library Edition, and use Stephen P. Morse's search tools to guide your search. Understand why you might find a record at one site and not another.
Research guides and tools