Many SEOs on forums have stated that they got out of the sandbox right after they got links from highly authoritative domains in their industry. Sometimes it was just one link; in other cases, it was several links.
From that we can conclude that the only way to speed up your site's exit from the sandbox is to get links from trusted domains.
Taking Advantage of Other Domains to Avoid the Sandbox
I have not tried this strategy, but it works according to some SEOs. If you want to launch a new website, but do not want to wait for the sandbox, then register a sub domain located on a trusted domain. New subdomains may avoid the sandbox altogether or have a considerably shorter stay. Just make sure the domain itself is good and trusted.
According to Rand of SEOMoz.org, you never "pop out" alone. It seems that Google has certain internally triggered events where a bundle of sites suffering from this issue all "emerge" to their expected rankings on the same day. You'll sometimes see forum threads and chatter about these "sandbox releases."
Another interesting statement: it seems you can lengthen your stay in the box by exclusively attracting the more typical "low quality" links that signal manual link building campaigns, such as built-for-SEO directories, article submission sites, reciprocal links, dofollow blog comments, forum signature links, etc. (this is speculation on my part, and more correlation than causation, IMO).
Also, once in the sandbox, avoid endless title, content and link tweaking. Just make sure you're not overoptimizing (titles too aligned with internal links, etc) and let Google do its thing.
The sandbox can drive you nuts if you don't have patience. Again, focus on site development during this period, or purchase AdWords and start conversion optimization, so when the site finally ranks, you will have a streamlined conversion path.
