Put your site on a unique IP, get a backup account at another host (I’d just go really cheap since it’s only a backup-but must have a dedicated IP as well), then you sign up for a failover service.
It’s only $14.95/year at dnsmadeeasy.com for the cheapest plan, which you can add failover to for $4.95.
Check out the plan comparison page.
The $59/year plan comes with 3 failovers, if you have a use for the extra resources/features.
Also, note that depending on your site and what’s going on there, you might not need a full up-to-date version of your site. If contact is what’s important, you could just have a maintenance page that says something like, “Our site is down for maintenance but should be up shortly. Please try again in a little bit, or use this contact form if you need to reach us immediately.” Or something like that.
The DNS site has more info on it and they can help you make sure it’s all setup correctly.
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Or save $97 with THEFULL97.