On the second day of Rosh Hashana, after reading about the akeda (and some other stuff), we read Yirmiya 31:1–19 as the haftara: why this haftara?
The Haftorah is very moving and probably hold the record for the most Jewish songs from one Haftorah! It especially relates to the Rosh Hashanah theme of Zichronos.
The Haftorah discusses God bringing the redemption, which may connect to the theme of Zichronos, as it involves God 'remembering' the Jews. Next is the scene of Rachel weeping and God promising redemption, which is part of the theme of Zichronos - recalling the Avos or Imahos as a merit to end the exile. It then describes Ephraim's Teshuvah, including the different parts of teshuvah, an important thing to mention during the Aseres Yemei Teshuvah: כִּי אַחֲרֵי שׁוּבִי נִחַמְתִּי וְאַחֲרֵי הִוָּדְעִי סָפַקְתִּי עַל יָרֵךְ, בֹּשְׁתִּי וְגַם נִכְלַמְתִּי כִּי נָשָׂאתִי חֶרְפַּת נְעוּרָי. Finally (as Rashi mentions) it cites God remembering Ephraim, which is one of the pesukim of Zichronos included in the Rosh Hashanah mussaf.