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I don't think the 7805 is usable for 6V input since it requires at least 7.2V at the input. This is because 7805 is not a low dropout regulator -- it is a conventional linear regulator using Darlington connected NPNs if I am not mistaken, so the voltage drop is more than 2 Vbe. I think it is actually 2Vbe + Vsat (for PNP driver).
It's too bad the original poster hasn't responded back. If this is a mains powered device then a LDO regulator with less than 1V DO is the way to go. If it is battery powered, then a swticher like I suggested is the right solution. With out any data it is impossible to know...
The 78xx data sheets have charts that show dropout based on load. The problem with taking one measurement is that the parts have a range of dropouts. Also, the point of regulation is to avoid load causing the voltage to fall. Might as well just use a diode to drop the voltage otherwise. The real problem is not the voltage level but the dropout transient which causes all sorts of problems. Dirty power is a recipe for problems.
Generally accepted engineering practice is to give at least 2V headroom for a 78xx Vreg. 6V to 5V is going to be extermely subject to load induced transients.
The DC-DC converter is basically the same as the switch mode design I pointed out. It's a lot cheaper to build your own but the DC-DC is a reasonable way to go if you don't might the cost.