Due to its importance and widespread use in industry, automated testing of REST APIs has attracted major interest from the research community in the last few years. However, most of the work in the literature has been focused on black-box fuzzing. Although existing fuzzers have been used to automatically find many faults in existing APIs, there are still several open research challenges that hinder the achievement of better results (e.g., in terms of code coverage and fault finding). For example, under-specified schemas are a major issue for black-box fuzzers. Currently, EvoMaster is the only existing tool that supports white-box fuzzing of REST APIs. In this paper, we provide a series of novel white-box heuristics, including for example how to deal with under-specified constrains in API schemas, as well as under-specified schemas in SQL databases. Our novel techniques are implemented as an extension to our open-source, search-based fuzzer EvoMaster. An empirical study on 14 APIs from the EMB corpus, plus one industrial API, shows clear improvements of the results in some of these APIs.