I have already conducted a study, however, after publication of this study, new question comes to my mind, can I do secondary analysis of this data to answer my new question?
Yes you can. But, since you have already done the analysis and have published your results, it would be better if you collect more data (if possible) and then continue with the analysis and justify it.
Maybe this could help:
Article Secondary analysis of existing data: opportunities and implementation
I agree with Hiran but would say that it is not a neccessary prerequisite to collect more data. In our field many researchers publish new analyses on existing datasets that result from questions that have arisen through previuos work and publications.
Having that said, I do think that the analyses should be somewhat innovative or additive to what you have done before and you should indicate that the study is a reanalysis of previously published data.