No, it is not necessary. Given that there is a test that accounts for heterogeneous variances (Welch's t-test), you can simply conduct it. For one, the tests for homogeneity of variance (HOV) are problematic in a number of ways. Some lack power, they - like other statistical tests - are too powerful with large sample sizes, effect sizes are missing for these tests, some are faulty under non-normality, ...
The typical approach for most applied researchers is to conduct Levene's test, then decide whether to conduct Student's t-test or Welch's t-test based on the result of Levene's test. However, Zimmerman (2004) showed through simulation that conditioning the test on the result of Levene's test distorts the p-value of the test i.e. your p-value from Student's or Welch's is not reliable when you choose which one to do based on Levene's test. Furthermore, given that Welch's test is almost as powerful as Student's test under HOV, and it is much more powerful when HOV is absent, it is advisable to "just do Welch's test".
Zimmerman, D. W. (2004). A note on preliminary tests of equality of variances. British Journal of Mathematical and Statistical Psychology, 57(1), 173–181. https://doi.org/10.1348/000711004849222
Here is another paper that gives the same basic advice:
Delacre, M., Lakens, D., & Leys, C. (2017). Why Psychologists Should by Default Use Welch’s t-test Instead of Student’s t-test. International Review of Social Psychology, 30(1), 92–101. https://doi.org/10.5334/irsp.82