The Messenger of Allah ﷺ liked to begin with the right side as far as possible in all conditions: in his purification, and combing. The narrator Muslim added: "in using tooth-stick," and he did not mention "in all his conditions".
كَانَ رَسُولُ اللَّهِ ﷺ يُحِبُّ التَّيَمُّنَ مَا اسْتَطَاعَ فِي شَأْنِهِ كُلِّهِ فِي طُهُورِهِ وَتَرَجُّلِهِ وَنَعْلِهِ قَالَ مُسْلِمٌ وَسِوَاكِهِ وَلَمْ يَذْكُرْ فِي شَأْنِهِ كُلِّهِ