The Targumim, the Talmud, Josephus, etc. Basically as commentary but even more. For example, if a NT writer uses an idiomatic phrase or other abstract form of speech that is unique to Judea or the ancient near east and is lost to us by time and geography, then seeing the same used in other places helps me better grasp the intent. Or, perhaps there are cultural debates happening that are mentioned that we are unaware of. If we are unaware, then when we read we have to fill in the blank with what we understand. For example....
Luke 14:1 Now it happened, as He went into the house of one of the rulers of the Pharisees to eat bread on the Sabbath, that they watched Him closely.
Can you tell me much about this verse? Is there anything deeper that might help me understand the concept of things he might say or do while there? The answer is yes. You see, in that day there were two schools of Pharisaical thought. One was called Beit Hillel (School or House of Hillel) which is where Paul attended by the way. I know that because I know that Gamaliel was Hillel's grandson and that he taught at his grandfather's school. Beit Hillel was known for teaching "the Spirit of the law." Now the second school was known as Beit Shammai, and it was known for teaching "the letter of the law." By the way, the influences of these two schools and them being defined by teaching the spirit or letter of the law... is the basis for Paul's "spirit or letter" points throughout his letters, But I digress.....
Anyway... because of the understanding of what was taught in each of the two schools, we can know that Jesus was eating at Beit Shammai. How do I know? Because of how they tried to set him up...
Luke 14:2 And behold, there was a certain man before Him who had dropsy. (3) And Jesus, answering, spoke to the lawyers
(Lawyers... not as in civil law, as in God's law) and Pharisees, saying, "Is it lawful to heal on the Sabbath?" (4) But they kept silent. And He took him and healed him, and let him go. (5) Then He answered them, saying, "Which of you, having a donkey or an ox that has fallen into a pit, will not immediately pull him out on the Sabbath day?" (6) And they could not answer Him regarding these things.
At Beit Shammai it was taught that no work of ANY KIND would be done on Shabbat. But at Beit Hillel, doing "good" on Shabbat should be expected. So, messiah was at Beit Shammai, the House or School of the sect of Pharisees that received the brunt of all of his rebukes and admonishments. In fact, as a side note... the School of Shammai received ALL of the correction save for on the topic of divorce. When he disagrees with the Pharisees on that one, he is disagreeing with those from Beit Hillel.
So not to be overlooked... Jesus went into the school of Pharisees that resisted him the most "and ate with them." We can learn a lot from that.
Blessings.
Ken