Yes I would create a search object that stores the terms of your search, something like

class SearchTerms
    {
        public string Term1;
        public string Term2;

        //etc.
    }

then on the search button click populate a new istance of this object and assign it to a session

SearchTerms st = new SearchTerms();
            st.Term1 = "abc";
            st.Term2 = "def";
            Session["SearchTerms"] = st;

then on page load do something like this

if (Session["SearchTerms"] != null)
            {
                SearchTerms st = (SearchTerms)Session["SearchTerms"];

                //use the st object to do your search
            }