The Poskim discuss whether it is permissible to order a non-Jewish taxi service to pick one up immediately after Shabbat. Obviously, the non-Jewish driver would have to violate the Shabbat in order to reach the address on time. The question is whether his driving on Shabbat is considered Melacha done for the Jew, which is prohibited to benefit from.
Rav Yishak Yaakov Weiss (1901-1989), in his Minhat Yishak (Vol. 6:25) was asked a similar question with regard to calling a non-Jewish driver on Friday to pick up Jews who spent Shabbat in Meron at the tomb of Rabbi Shimon Bar Yochai. He concluded that it is prohibited to do so.
This strict ruling was questioned by other authorities, including Rav Shelomo Miller. They rule that it is permitted based on the Mishna Berura (Rav Yisrael Meir Kagan, 1839-1933, 276:27). The Mishna Berura cites the opinion of the Taz (Rabbi David Segal, Poland, 1586-1667) who permits instructing a non-Jewish maid to wash dishes, which is intrinsically a permitted act, even though she will have to turn on the lights in the kitchen to do so. He reasons that the Jew has no direct benefit from the light; it was her prerogative to turn them on, and she did so for her own benefit. The Mishna Berura applies this leniency to sending a non-Jewish messenger on a mission on Shabbat, even though he will kindle a candle to illuminate his path. Based on these precedents, these Poskim are lenient and permit ordering the taxi for Mosei Shabbat, since the Jew did not direct him to drive on Shabbat; he is doing so for his own benefit.
This lenient opinion is the accepted Halacha. However, one must exercise caution not to accidentally order a Jewish driver.
It is possible that the Minhat Yishak was strict on this issue, because he was dealing with a case where the taxi company charged for the drive to the Jew on Shabbat. In such a case, the time driving on Shabbat constitutes an integral part of the service provided to the Jew.
SUMMARY: It is permitted on Friday to order a non-Jewish driver to pick one up immediately after Shabbat.