The second illegal immigrants most likely came during the period after 1875, after the Page Act was passed. The Page Act essentially forbade the involuntary importation of individuals (namely for the purposes of prostitution) or the involuntary importation of any individual from China, Japan, or elsewhere in Asia, illegal. This law was essentially a response to those pro-immigration representatives from the South who saw Chinese immigration as an opportunity to essentially enslave a new class of indentured servants. This law and the rhetorica around it eventually led to the Chinese Exclusion Acts in the 1882 (amended in 1884), 1892, and 1902. (The laws were 10-year exclusions with sunset triggers-- the third version passed in 1902 took out the sunset.)
Essentially, although numerical data are scarce, the second group of illegals were probably Chinese women smuggled here. I find this interesting because this raises an important about women and "sex trade" with respect to globalization that we struggle with today, but it also reminds us that these sins are not new ones. They are, in fact the oldest sins in The Book. Doing away with all trade or all commerce because some commerce is "sinful" would be to cut off your nose to spite your face.