People who are working on reducing their accents and who want to sound more like native speakers must also continually work on improving their overall language fluency.
Never neglect other important skill areas of English. Make sure that you are constantly learning new vocabulary and trying to express yourself better in general. Look up any new words that you don’t know and try to use them right away so that they will become a part of your new speech. Also, fix those final grammar mistakes that even the most advanced learners of English sometimes continue to make.
Become a perfectionist about all aspects of your speech.
Here is why:
Native speakers may think they “hear” an accent when in fact they hear a grammar error or the wrong usage of a word. They don’t have the depth of experience to analyze the type of error they heard. It will simply sound “foreign” to their ear and your accent will suddenly appear stronger to them than it really is.
Remember, fluency means expressing yourself well in all aspects of language: vocabulary, grammar, accent, writing, spelling, idiomatic expressions etc.
Work on all of these areas. Which ones have you been neglecting, and what are you going to do about it?