Monopolies don't maximize welfare because they set prices above the equilibrium price, leading to dead-weight loss. It is possible for the government to provide a per unit subsidy to a monopoly until the producer's marginal cost equals the consumer's marginal benefit at the monopoly's chosen quantity, which would maximize welfare. This change would increase producer surplus and consumer surplus in this market.
To a monopolist producer, a per unit subsidy is essentially equivalent to shifting the demand curve up by the value of the subsidy. It would be possible to shift the demand curve up until the optimal quantity produced is the higher equilibrium quantity. This requires that the subsidy provider knows the complete demand curve to set the optimal subsidy.
The subsidy must come from somewhere, like a tax on a different market. It seems plausible that there would be situations where even if all tax incidence fell on consumers, consumer surplus would still be greater after the subsidization. If consumers and producers are overall better off after a subsidy, why don't consumers do this without government intervention?