I think Picos de Europa would be a good option in June, specially as it's close to Santander. Not comparable to Extramadura for birding but excellent for flowers, butterflies and walking at that time of year.

Dear Victoria,
What a great opportunity to discover the unknown northern plateau. Mid June temperatures is perfect for making daytrips and enjoy the beautiful landscapes of Soria.
We offer lots of nice walking daytrips in which you can find birds, flowers and butterflies!
Look at our website to find more information.
Best regards, Conny