By all reports, the 1870 federal census was a mess. It was the last census for which U.S. Marshals were used as census takers. As there weren’t enough Marshals, a number of men were made Assistant Marshals for the purpose, many of whom were Northerners who were said to be only semi-literate. Their lack of qualifications were blamed for leading to errors and fraud, ultimately culminating in undercounting the South by an estimated 10%.
In the North, three cities thought themselves significantly undercounted. They also complained that the census takers had not recorded street addresses, leaving no way to confirm that all addresses and families had been recorded. Their complaints were loud enough to elicit an order from President Grant to conduct a second enumeration for New York City, Philadelphia, and Indianapolis, the first, second, and twenty-seventh largest cities in the nation in 1870. For those three cities, both enumerations were retained and are available for research.
Take, for example, Sarah Wilkins of Philadelphia. She was first enumerated 22 June 1870, when she was located in the 4th Ward. No more specific location is given for her. She was 54, black, had $6,000 in real estate and $700 in personal estate. Included in her household were Sarah, 14, mulatto, and Mary, 18, mulatto. All three were hairdressers born in Pennsylvania.
Sarah Wilkins was enumerated again 15 November 1870, when she was located more specifically at 610 Pine Street in the 14th Division of the 5th Ward. She was 60 and colored. Included in her (apparent) household was only Sarah, 15, colored. Nine others were enumerated at the same address, but this enumeration did not separate multiple families living within a single dwelling. It also did not include personal information other than name, age, race, and gender. In New York City, they added occupation and birth place to the second enumeration, but no more.
Each enumeration has its strength in terms of genealogically helpful information, but using both enumerations jointly gives the most complete picture. Finding both, however, can sometimes be a challenge.
Unfortunately, FamilySearch doesn’t catalog the enumerations separately. None of their entries include an indication of first or second enumerations. Browsing the 1870 census will allow you to browse by wards, but FamilySearch includes both (unidentified) enumerations in a single link for each ward. You have to differentiate for yourself.
Ancestry, on the other hand, differentiates for the user. Browsing the 1870 census for New York City (which then was only what is now Manhattan), separate sections are titled, for example, “New York Ward 01 District 01” and “New York Ward 01 District 01 (2nd Enum).” The same is true for Philadelphia and Indianapolis.
Ancestry also appears to have indexed more completely than FamilySearch, but as in many other instances, you may need to try both for greatest success. If you don’t find your subject in the indexes for both enumerations, you may need to browse the rolls individually. While FamilySearch doesn’t differentiate between the two enumerations in their catalog, they do have them, so they are viewable there as well.
The National Archives’s microfilms of the 1870 federal census population schedules are in RG 29, Records of the Bureau of the Census, microfilm publication M593. The pertinent rolls (and their Family History Library microfilm equivalents) are listed below. Using these roll numbers, you can identify second enumerations on FamilySearch.
New York City, first enumeration, rolls 975–1013 (FHL #552,474–552,512)
New York City, second enumeration, rolls 1014–1053 (FHL #552,513–552,552)
Philadelphia, first enumeration, rolls 1387–1414, 1445 (FHL #552,886–552,913)
Philadelphia, second enumeration, rolls 1415–1444 (FHL #552,914–552,943)
Indianapolis (Marion County), first enumeration, rolls 340–341 (FHL #545,839–545,840)
Indianapolis (Marion County), second enumeration, rolls 338–339 (FHL #545,837–545,838)
For New York City in specific, researchers should know that boundaries for wards and districts changed between the two enumerations, which can make finding your subject more challenging if they are not found in indexes. To this end, Steve Morse has created a useful tool to translate street addresses into enumeration districts.
Take a look back at your research. If your subject is in one of these three cities in 1870, have you found both enumerations for them? If not, some additional information, perhaps critically important, may be waiting there for you.
 Joel Weintraub, “Second Enumeration of 1870 Census,” Stevemorse.org (https://stevemorse.org/census/1870secondenumeration.html : 21 April 2020); Mary Kircher Roddy, “1870 Second Enumeration,” mkrgenealogy.com (https://www.mkrgenealogy.com/searching-for-stories-blog/1870-second-enumeration : 21 April 2020); The Ancestry Insider, “Darned Second Enumerations,” ancestryinsider.org (http://www.ancestryinsider.org/2011/09/darned-second-enumerations.html: 21 April 2020). The Census Bureau is peculiarly silent on the matter (https://www.census.gov/history/www/through_the_decades/overview/1870.html : 21 April 2020).
 Sarah Wilkins household, 1870 U.S. Census, Philadelphia, 4th Ward, Philadelphia County, Pennsylvania, 1st enumeration, 22 June, page 33, stamped 268[A], dwelling 211, family 315 (NARA M593, roll 1390).
 Sarah Wilkins household, 1870 U.S. Census, Philadelphia, 5th Ward, 14th Division, Pine Street, Philadelphia County, Pennsylvania, 2nd enumeration, 15 November, page 2, stamped 107[B], #610 (NARA M593, roll 1419).
 NARA Series to FHL Film Conversion, M542–M694, FamilySearch, (https://www.familysearch.org/wiki/en/NARA_Series_to_FHL_film_Conversion,_M542-M694 : 22 April 2020).