Only between 10 and 15% of the official lists for assisted immigrants arriving in South Australia in the 19th century under the sponsorship of the Colonisation Commission have survived. Also search Trove for newspaper articles announcing arrivals.
Not all 19th century travellers sailed directly to Adelaide. In the early 1850s many ships sailed directly to Melbourne due to the attraction of the goldfields so many immigrants came to Adelaide via coastal vessels. The Public Record Office of Victoria has Melbourne passenger lists searchable online.
A search of Trove may reveal the person you are researching in newspaper articles listing arrival/departures.
The State Library also has specialised indexes of newspaper articles listing passenger arrivals and departures between Australian Colonies which may reveal names that Trove has not recognised.