In my past job I was building online banking for a small Los Angeles based bank. I'm pretty sure they offer the service you seek. The online banking service I was helping build is still in beta but I think you can request to be added to the beta. Try City National Bank (cnb.com).

Is that really the standard in the US? In Lloyds (UK) same-bank transfers are free and instantaneous, first time requires two-factor auth (mobile call confirmation). It works pretty well actually. Web / mobile enabled. I haven't seen a check in years...