As @EthanBolker said, a good literature search is an iterative process that evolves along with your research progress. The very notion of a "perfect search string" from the very beginning constrains you to think only in terms of what you knew when you started the project without adjusting for your continuous learning as you advance in the project.
I would further add that it is a widely mistaken concept that the primary purpose of the literature review of a scholarly article (which I call a "background literature review") is to summarize the related literature. Well, that is often accurate in practice, but in my opinion, that is not the most useful thing that it might do. Rather, the primary purpose of a background literature review should be to clearly carve out the novel scholarly contribution of your article in relation to the literature. That is, by the time you have completed the article, the background literature review should focus on clearly showing how the existing literature is related or similar to what you are doing, but how your article goes beyond the existing literature and makes novel scholarly contributions beyond what exists.
You cannot write such an effective background literature review at the beginning of your project when you yourself are not quite clear what is the extent of your contribution. Although it is certainly good to do an initial search before you start so that you have an idea of the research landscape, it is only by the time you are finished and you can properly appraise what you have done that you would be able to complete a background literature review that properly places your contribution in context of all relevant literature. So, from that perspective, there cannot be a meaningful notion of a "perfect search string" at the beginning of a research project.
I think what you have described is a good way to start: "I try to find an initial paper that is relevant and the browse through its cited papers to see if there is any paper that describes what I am looking for in a better way." In addition, you should probably use Google Scholar to do a forward citation search, that is, find articles that have cited your initial papers and see if something more recent is related. Then, after you have completed your project and have a good idea of what your contributions are, you can try to search on the keywords based on your contributions to find if anyone has already done anything similar. Then read such articles to clearly understand how your contributions are different so that you can then write a background literature review that clearly highlights the uniqueness of your contributions.