A Person Who Exceeds the 2000 Amah Limit, But is Permitted to do so — Eruvin 44b
תלמוד בבלי מסכת עירובין דף מד/ב
מי שיצא ברשות ואמרו לו כבר נעשה מעשה יש לו אלפים אמה לכל רוח
Chazal allowed certain people to exceed the 2000 amah limit of Techum Shabbos. According to Rashi here, this includes witnesses who are on their way to testify concerning the new moon, to save a person from bandits or from drowning, or a midwife who is going to assist a woman in labor.
In Sefer HaPardes, Rashi adds that it is also permissible to go beyond the techum in order to save a Sefer Torah. R' Yosef Engel posits that if it is permissible to go beyond the techum in order to save a Sefer Torah, it is also permitted, once the person reaches his destination, to go 2000 amos in all directions from that place.
However, notes R' Shlomo Zalman Auerbach, the basis of this leniency is the Rabbis concern that if they do not allow the person some leeway, then next time a similar situation arises, around he will avoid it (see Mishnah Berurah 407:3). Hence, they could only permit this leniency in regard to their own decree. Hence, since most Rishonim are of the opinion that there is a D'Orysa, Torah law prohibition to exceed 12 mil (2000 amos x 12; see Mishnah Berurah 404:7), the Rabbis cannot extend the techum beyond that limit. Thus, if a person had to travel for one of the permitted reasons and traversed 11.5 mil to get to his destination, he can only walk the additional 500 amos to the 12 mil limit, and no more (Nishmas Avraham, Orach Chaim 507:1 — see there that in the end R' Shlomo Zalman is inclined to give even the person in this situation 2000 amos from his destination).