Ask HN: Any tools to force TLS handshake to use “more secure” network?

Hey I'm not a security engineer but it occurred to me that when using an unsecured public WIFI network the real concern is in doing the initial TLS handshake over that network. My question is - If I switch my laptop to use a tethered network via my cellphone prior to hitting an HTTPS website and then switch it back to using the public WIFI after the connection has been established would I be improving my security while gaining the benefit of higher speed and cheaper wifi access (as opposed to doing the whole thing via the tethered connection)? If so are there any tools to handle this automatically?