What are a beneficiary’s rights to an inheritance if they die right after the person that they inherited from?

In some limited situations in Texas, a person is treated as having predeceased the testator if the person doesn't survive for more than a few days after the testator dies. In this case, your mother survived for a month after her sister, so unless her sister's will set other terms, your mother inherited from her sister.

The question of whether everything is part of your mother's will is probably a separate issue. Your mother evidently didn't enact a new will after she inherited from her sister, but depending on what kinds of property your mother inherited and the language of your mother's will, her will would still apply to the property she inherited from her sister.

Think about it like this. If a testator makes out a will stating who gets what proportions of his property, and uses language that makes it clear that it applies to property the testator might come by in the future, he doesn't have to make a new will every time he deposits a check into the bank account, acquires a new property, etc. But if the testator wrote a will that said only "my house on 22d avenue goes to Bob," and then won the lottery, sold that house, and lived in a rented apartment for the rest of his life, and died leaving millions in assets, the will really wouldn't tell us much, would it.