The union-closed sets conjecture states that, in any nonempty union-closed familyFof subsets of a finite set, there exists an element contained in at least a proportion1/2of the sets ofF. Using an information-theoretic method, Gilmer recently showed that there exists an element contained in at least a proportion0.01of the sets of suchF. He conjectured that their technique can be pushed to the constant3−52which was subsequently confirmed by several researchers including Sawin. Furthermore, Sawin also showed that Gilmer’s technique can be improved to obtain a bound better than3−52but this new bound was not explicitly given by Sawin. This paper further improves Gilmer’s technique to derive new bounds in the optimization form for the union-closed sets conjecture. These bounds include Sawin’s improvement as a special case. By providing cardinality bounds on auxiliary random variables, we make Sawin’s improvement computable and then evaluate it numerically, which yields a bound approximately0.38234, slightly better than3−52≈0.38197.