West - Intl-Dom PHX Connection: Staying Indoors?

I have a trip next month that will make me do LHR-PHX-ONT with BA and then connecting to Southwest .

I see that I will remain in Terminal 4 for the BA-Southwest connection. My question is after going through immigration and collecting my bags, is there an indoor path where I can go to the Southwest counters to re-check my luggages without having to step outdoors? Reason I am bringing this up is because I ABSOLUTELY HATE THE DESERT HEAT. I do not want to expose myself to the 105F+ temperatures nor those hot desert winds blowing at me. It's bad enough that I'll have to endure some of those at the air bridge. So if I can stay in an indoor-air conditioned facility throughout the transit process, It'll be terrific.

It has been ages since I last flew via PHX so I have almost no memories of the terminal layout.

BA and WN are both in the Barry Goldwater Terminal 4, so unless there is some other airline involved, you will have no need to go outside on your transfer through Sky Harbor.

You know usually in September tempuratures only make it to 95-100F :D

Like wilbur said though, you will have no cause to go outside and expose yourself to our awful awful warmth. Plus, the BA flight gets in later in the day, not QUITE so much sunshine then ;)

One tip having flown this route last week- the FA in flight announced that if your are connecting on Southwest, you can't use the check in desk right outside of customs/ immigration, but you would have to lug your bag to the WN check-in.

Doesn't apply to me since I live in PHX but hope this helps...