The Italian team said in a statement on Friday that Adami would manage the Driver Academy and testing of the previous cars programme.
Ferrari, who finished fourth overall last season without winning a race, said Hamilton's new engineer would be announced in due course.
Hamilton, who joined Ferrari last season from Mercedes, dismissed as "just noise" reports he and Adami were not getting along after terse radio exchanges between the pair from the opening race in Australia.
"Do we have disagreements? Yes, like everyone does in relationships. But we work through them. We are both in it together," the 41-year-old said last May.
The awkward exchanges continued right to the end of the campaign in December, however.
Hamilton failed to stand on the podium all season, unlike teammate Charles Leclerc, in a career low for the Briton.
Adami had previously worked with now-retired four-time world champion Sebastian Vettel and Spaniard Carlos Sainz at Ferrari and has been with the Maranello team for 11 years.
Ferrari are due to hold a livery launch next week before the 11 teams test behind closed doors in Barcelona from January 26th-30th.
The 24-race season starts in Australia on March 8th, with the sport entering a new engine era in a major shakeup of the technical rules.