Did a quick run through the rest of hamartano in the NT, namely, in the gospels, Acts, general epistles, and Revelation. The same finding applies. To sin is to do wrong. I didn't find the Jewish Law so much as an explicit standard in these other books. In the gospels, sinning is often "against" someone or something. There are sins against a brother (e.g., Matt. 18:21). There are sins against a father (Luke 15:21), against heaven (Luke 15:21), against the temple and Caesar (Acts 25:8).
The notion of sinning against heaven is interesting. I suppose one might say that all sins are sins against heaven. But it's important to recognize that this is a theological idea I am having right now--the biblical authors might affirm this idea if you asked them, but it is generally not something their texts have in mind in the original contexts. [it is thus overreading the texts to ascribe this as an element of the original meaning of these passages--the overload fallacy]
The vast majority of instances in the rest of the NT where hamartia is used also refer to wrongdoing, wrong that has been done or wrong that might be done. In my quick survey, I did not notice any instance in the Synoptic Gospels and Acts where this was not the basic sense. John, written after Paul and later in the first century, had a couple instances where Sin as a power seemed to be in view (John 8:34; 9:34) and one that seemed to have the sense of guilt (9:41).
Hebrews of course brings in strongly the sense of sin as causing impurity, but this is a consequence of sin, like death, rather than part of the core definition. The word agnoema, "act of ignorance," in my opinion, means more than the OT does in relation to this word. I believe the author refers to sins the audience committed before they came to Christ and were not yet enlightened, similar to the acts of ignorance in Paul's sermon in Acts 17.
James 4:17 speaks of sins of omission, but the same basic sense of wrongdoing is there, as with all the other references in the General Epistles and Revelation. I found four references in all to the sinlessness of Jesus--2 Cor. 5:21; Heb. 4:15; 1 Pet. 2:22; and 1 John 3:5. 1 John 5:16 also clearly points to different levels of sin.