Art Scroll siddur (Ashkenaz / Zichron Re'uven) implies that there should be a piyut before Unetaneh Tokef (It doesn't explain why this is needed, though). On the first day, there are already several piyutim there. However, on the second day, there is nothing between Avot and Unetaneh Tokef. Thus, some minhagim, move Lekel Orech Din to Musaph.

Art Scroll siddur (Ashkenaz / Zichron Re'uven) implies that there should be a piyut before Unetaneh Tokef (It doesn't explain why this is needed, though). On the first day, there are already several piyutim there. However, on the second day, there is nothing between Avot and Unetaneh Tokef. Thus, some minhagim, move Lekel Orech Din to Musaph.

See this related question.