How was the person for the second Tamid selected? Was there a second lottery?
The gemara on Yoma 26a explains that there was just one lottery that covered both services:
א"ר יוחנן אין מפייסין על תמיד של בין הערבים אלא כהן שזכה בו בשחרית זוכה בו ערבית
Rabbi Yoḥanan said: They did not hold a separate lottery for the slaughtering and sacrifice of the daily afternoon offering. Rather, the same priest who won a particular privilege for the morning offering wins the privilege for the corresponding task in the evening, i.e., for the afternoon service.
In this way, the morning lottery covered both services.