I think the cold is going to let up the first few days in the New Year.
I think the cold is going to loosen its grip from the first to the third of January.
「寒さが緩む」は to let up や to loosen its grip などを使って表現できます。
例：
ーI think the cold is going to let up the first few days in the New Year.
「お正月の数日は寒さが緩むと思います」
ーI think the cold is going to loosen its grip from the first to the third of January.
「1月1日から3日は寒さが緩むと思います」
ご参考まで！
カナダ 
