Say you have a name that ends in S, like Millis. If Ms. Millis owns something, like a book, is it Ms. Millis’ book or Ms. Millis’s book?
The short answer is “consult your style guide.”
Style guides exist precisely to make a ruling on issues like this, where there’s more than one acceptable answer and someone needs to pick one so everyone can just get on with their lives. If you don’t have a style guide, that’s fine: As long as you’re consistent, it doesn’t matter which rule you choose to follow in ambiguous matters like these, so just pick one that pleases you.
The long answer is that there’s a few schools of thought here. There are some style guides at each extreme—never add another S or always add another S—and both camps prioritize ease of remembering the rule over what the sentence might sound like if read aloud. That’s totally fine, and people will understand what you mean either way, so if you just want a rule you never have to think about, go with one of those. Personally, if I had to pick between these two, I’d say always add an apostrophe-S because I think the visual makes it more obvious to the reader.
Some style guides advocate a more nuanced approach, which makes the text easier to read aloud or attempts to varnish over a fundamentally arbitrary convention with something that looks vaguely like logic. The Chicago Manual of Style, for example, once had one set of rules for your average, everyday proper names, but a different set for historical figures, so it would be “Ms. Millis’s book” but “Moses’ book.” That book has since since simplified its rules considerably, to the relief of many.
I like an approach with a little nuance here, and my personal favorite is the one advocated in the house style guide at my current employer: Use apostrophe-S if the result is pronounceable; if it isn’t, just add an apostrophe. This, I think, makes the text easy to read without requiring editors to memorize a list of exceptions. It may result in occasional variations between editors, but readers are unlikely to notice.*
One important exception: When the proper noun ends in S because it’s plural, just add an apostrophe, like you would for any other plural noun. So it would be “the Joneses’ house” or “the Smiths’ garden.”
*Any who do notice should reward themselves with a bit of chocolate and a moment of silent self-congratulation. Letters to the editor on topics such as these go straight into the circular file.