We consider the mesorah continuing past the mishnah through the gemarah and then continuing through the geonim, the rishonim (such as Rashi and Rambam, etc.), and then through the acharonim such as the Shulchan Aruch, the Mishna Brurah and modern poskim such as Rav Moshe Feinstein, Rav Elyashiv, Rav Ovadiah Yosef, Rav Dovid Feinstein, Rav Chaim Kanievsky, my LOR, etc.
Thus even though we do not have semicha, we still have the chain of mesorah recognized as an unbroken chain. Of course there are different groups that may have different minhagim (such as sfardim and ashkenazim) but in all cases it is still an unbroken chain of mesora.
Each person can look to his LOR and trace back the chain of mesorah that he has to the common point.
There are those who have tried to reestablish semicha (based on the Rambam) but so far we have not been able to get everyone to agree (one of the Rambam's requirements).
See the discussions on eating turkey as an example as to how mesorah is considered. This applies even though the question arose long after semicha ended, showing that we still consider the mesorah as continuing. Within this discussion, you will see that since birds are defined as not kosher based on a list, we consider that one should follow the mesorah as to which birds are kosher and which birds are not. The rabbonim discuss whether or not one can derive simanim as to whether or not an unknown bird is kosher (similar to fins and scales with fish or hoove and cud with animals) but most discussions nowadays rely on mesorah. That is too long a matter to go into here and can be found in other answers.