The title pretty much sums up the question.
If a question has already an accepted answer, should I still try to answer it?
Provided that the answer is not similar to the accepted answer, but an alternative method which may also solve the problem.
If you have something new to offer that answers the question and is not appropriate as an edit to another answer, then you are helping the community (and sometimes even the OP) by posting it as a new answer.
Sure, why not, if you think it's useful then do it!
(Though if it's just slight changes you might just want to submit an edit to the accepted answer).
Yes, If you think, the new answer will help future visitor, you should. It is encouraged to answer. More option is always better for All.
The answers here are not for the OP only, these are for all the visitors. It is also possible that, the accepted method may not work for some people while, your method is working for them.
And You will find many example of this situation, where a later answer has far many votes than the accepted answer.
If you are able to apply a different perspective to the answer or add to that answer a proven (real) salvation then, yes. Tested answers are encouraged. Opinions are not.