A gas turbine engine comprises a compressor section and a turbine section, with the turbine section having a first stage blade row and a downstream blade row. A higher pressure tap is tapped from a higher pressure first location in the compressor. A lower pressure tap is tapped from a lower pressure location in the compressor which is at a lower pressure than the first location. The higher pressure tap passes through a heat exchanger, and then is delivered to cool the first stage blade row in the turbine section. The lower pressure tap is delivered to at least partially cool the downstream blade row.