You need to look a little deeper into how Regulators work. A standard or switching regulator is not going to work. A 5V regulator as a dropout voltage at around 1.3V. This means that when you get within 1.3V of the target voltage, the regulator will stop working. Since 6V is not greater than 5V + 1.3V, it will not work.
This is one of the reasons that a 9V battery is not a good choice for the Arduino. Once you get to about 7V it no longer works making for some really poor utilization of the battery.
You will need to use a LDO regulator as a worse case option or a boost convertor as the best option.