Tweaking the new "title-tag"

I have selected to use a static page for my home page.

The old way of wp_title() used to format the title tab of the browser on the home page like so:

Site Title | Site Tagline

The new way of doing title tabs in the functions.php gives this for *every page*

Page Name – Site Title

I understand that for SEO purposes having the Page Title first before the Site Title may be better, but is there a way for the Home Page (ie http://treesourcetexas.com) still to show the Site Title first?

Additionally, I haven’t even gotten around to updating my other sites because of this whole wp_title() fiasco issue. I know that I have some pages on other sites where I’ve given a page a “title” only so I can identify it on the backend. But on the front, the template that the page uses does not show the Page Title at all.

So I suppose is there a way to tweak the whole “title-tag” deal?

Thanks