BOSTON — State lawmakers want more oversight of sheriffs’ budgets amid concerns about overspending, but the overseers of county jails say their costs are going up because Beacon Hill keeps piling on mandates like free phone calls for prisoners.

On Thursday, the state Senate approved a $2.3 billion supplemental budget to close out the previous fiscal year’s books by providing more money to cover rising health care costs, education and the state’s life sciences industry.

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