Even better is embedding the SCTs in the x509 structure itself so that you don't have to rely on obtaining/caching and the sending in the handshake. (Yes, there's some cases where a policy change my require the addition of additional SCTs—or different ones altogether—but this should be the exception not the norm.)
The web server sends a Signed Certificate Timestamp in the TLS Handshake¹. The browser will check that.
Apache support is coming², and other web server vendors are probably working on it as well.
¹https://tools.ietf.org/html/rfc6962#page-13
²https://httpd.apache.org/docs/trunk/mod/mod_ssl_ct.html