A simple rule of thumb: if you use "already" about something that has already happened, always use the perfect tense

"We have already finished"
"He said that they had already finished"

It is possible to form sentences where the straighforward past tense is correct, but you will be on much surer ground if you avoid using "already" with such tenses. The same guidelines apply to "yet" in a negative sentence:

"We haven't had our dinner yet"
"They said that they hadn't had their dinner yet"

If you don't observe this rule, you risk sounding like an uneducated native of Brooklyn in New York - surely a fate you would wish to avoid!