In this Academia piece, from 2010, a number of scholars answer three questions about the "New Search."
The first question: Is this phase over? My own answer, riffing on one of the commenters, is, it should be considered on pause, until more digestion of the sociological elements of Judaism(s) at the time of Jesus, along with broader culture at this time, is complete.
The second question asks each to comment on that social and cultural background. All stress Jesus' Jewishness while also showing that, in terms of education, style of life, teaching focus and many other things, they can still have large differences in interpretation.
The third question spills from this and asks about Jesus' originality.
Some of their specifics are interesting.
For example, John Dart, after talking about images of bad faith in Mark in general, notes the presumably original ending of the gospel is one more example of bad faith, as the two Marys run away from the empty tomb in fear, just as fear motivated Peter's denials, etc.
Dart also makes an argument of the early date of several non-canonical writings.
Several respondents talk about Jesus and halakhah. Several of them go beyond the basics to talk about how this influenced Jesus' understanding of purity issues, his disputes over this with the Pharisees and possibly others, and from that, his relations to the emerging, but not established, oral Torah.
Several place this in the larger context of "Hellenization" of Galilee. From a couple of books I've read on the issue, like Lee Levine, "Hellenization," even before the Maccabees may have "de-Hellenized" Galilee in some ways, was not in direct opposition to "Judaization" or whatever term should be used.
Many contributors note the great variety of Judaism(s) at this time. I think one thing that unites "New Search" scholars is going beyond Josephus' "four sects," way beyond, in some cases.
Richard Horsley notes that at least parts of what eventually became collected into I Enoch may have been considered "canonical" in some circles at this time. He argues for a sociological reinterpretation of the Daniel 7 vision.
It's also Horsley who talks about the low level of literacy in Judea and Galilee of this time in general.
Somewhat riffing on that, and the Lukan special material, Rainer Reisner argues for a relatively high educational level of Jesus.
Paolo Sacchi talks more about the varieties of Judaisms. He also says he doesn't think Jesus had that much focus on purity issues.
