I have tried different induction temperature and IPTG concentrations but nothing is helping to increase the solubility of the protein. the protein has 10x-his tag.
I'm not an expert about zinc finger, but if i remember well in this protein zinc play a structural role and is not so simple in the citoplasmic espression in E.coli have enough zinc avialable and more over even the presence of the his-tag may compete with the binding of the zinc.
In the past i worked in the recombinant production of Cu,Zn SOD1 those had similar issue since the zinc is structural and we solved it by performing periplasmic expression of the protein coupled with LB supplementation with Cu and Zinc solutions.
It may work even if your case? Nothing is sure with proteins but it something not so hard to try.
You can simply;
1) a leader peptide (pelB or OmpA) to your clone using PIPE cloning (https://proteocool.blogspot.com/2021/09/tips-of-week12-rapid.html)
2) Supplement the LB media with 1mM of Zinc cloride and Copper cloride
3) Perform periplamic extraction using osmotic shock
i suggest also to remove his tag, which may bind zinc, produce the native protein or a version carring different tag, eg strep tag.