Generally speaking, Tom does not try to keep his affair with Myrtle a secret. In Chapter One, for instance, Jordan tells Nick that "everybody" knows that Tom has a "woman in New York." She even calls the Buchanan house and this is how Daisy finds out that Tom is having an affair. In the next chapter, Tom takes Nick to meet her and makes no qualms about his secret life.
In contrast, it is the identity of his mistress which Tom tries to keep secret. Tom never tells Daisy or Wilson, for example, that Myrtle is the object of his affection. Moreover, in Chapter Two, he and Myrtle ride in separate train cars on the way to New York. Tom claims that this is to protect the "sensibilities" of any passengers from East Egg but, arguably, he does this because he wants to protect Myrtle from being gossiped about.
That Tom does not try to keep his affair a secret reveals the arrogant side of his character. His only interest lies in pleasing himself, instead of protecting the people around him.