Each state has different rules. Where I live, lawyers bid for a defender contract and then take all conflict work- cases PD can't handle - that comes their way in a given year. The contract is not set per case, and the special PD then hires its own lawyers as employees, but the State also pays the administrative costs of running what is essentially another state agency.
Other states pay each individual lawyer per case, and murder cases get quite pricey. Even high paid defender can be among the pool of potential lawyers, and the State pays them an hourly rate, one usually much, much higher than the going PD rate. One capital murder case in Seattle resulted in over 6 million dollars in special defender fees and costs a couple of years back. Although CT does not have the death penalty, this case is still going to be very expensive no matter what.
