Get some jerry cans.

300 miles on a tank is OK. You have to pass emissions testing so radical changes to your powertrain are likely to be expensive (heck, they're not cheap anyway). Without knowing your specifcs it's hard to say, but it is possible that you could gain some economy with a re-programing of your engine computer (or replacement) but there are still mechanical limits to what that can accomplish (valve size, lift, duration, & timing, compression ratio, internal efficiency of things - like mains - etc etc etc).

With really careful driving you might be able to squeeze another 100 miles out of a tank - being in overdrive but below 45 mph is a place to start if you're on the level. A dashboard vacuum gauge can help if you know how to interpret it in various load conditions. But in a bugout situation I'm betting that driving demands are going to kick one's fuel milage severely in the opposite direction - and even a short term in 4wd will suck some serious fuel.

By far the cheapest thing to do is add capacity. Safest way GENERALLY is to add a fuel tank (if there are any aftermarket ones for your S-10 Blazer). Cheapest way is what's already been suggested - jerry cans. It takes discipline to keep them fueled with stabilizer added (and you still should empty/refill often - ideally monthly, especially if, like here, fuel blend varies with the season).

[rant] Unlike many other countries, no one here sells a good little 4wd with a small 4 cyl diesel (The Liberty 2.8L is a maybe, I guess). And now too many companies have replaced good little 4 cyl gasoline chuggers with multi-valve high rpm engines that truly suck at motoring along whilst sipping fuel. Ironic, since the tech now really would allow far superior small truck engines compared to the older carburated non-EEC engines. Everyone wants to blast down the highway instead... that takes HP and rpm (and fuel). Some of the same small trucks sold here are available elsewhere with neat little diesels; I've used them, and they are more suitable for this sort of thing. Moving to bigger vehicles like I drive, I SERIOUSLY object to the price premium I have to pay for a diesel engine - I simply cannot justify the cost difference; I will never recoup the savings in fuel. [rant]

Assuming that changing vehicles is not an option, planning and prep and practice (all of which CAN be entertaining if you wish) will be more important than your range. I'm oversimplifying, but we have always planned 3 stages (the intermediate stage depends on bridges, so maybe it's 4 stages). Vehicles, AT bikes (or canoes), and on foot (or snowshoe). In answer to another question, all the ATV's I've ever owned or used will run without problem far longer than I care to stay astride one, in the worst conditions imaginable. But I'm not equipping my family of 6 with ATVs - I no longer own even one as of last month.

Tom