If i can suggest you a thing, for currents at the limit of the regulator, use two of them in parallel (LM317) ..... but, is important, try to get two from the same lot of production, so the characteristics are the more identical possible, included the tolerance shifts, and they does not interfer one with the other (using components from different lots of productions, that can have opposite tolerance shifting, and they don't work good in direct paralleled configuration)
The lot of production, same as country of production, is usually indicated from the other numbers on the component, the ones under or around the "LM317" mark ..... if you get two that have the same, identical numbers and letters, you're ok.
Using them in direct parallel, keep all the connections shortest as possible, do very good soldering, and still use heatsink (you don't need to insulate the regulators one from the other, cause tabs are in common, but you still need to insulate them from case, if the case is connected to power)
Anyway, i've tried on-the-fly for check if it work, before post, but only with 2 in parallel, cause i have only 2 of the same production lot, all the others are different and not risked ..... so i cannot say if 3 or more can work in the same way ..... with two, and a 0,56 ohm constantane resistor, it feeded a car lamp at 2,2 A for 10 minutes, becoming hot, but not "egg-cooking" hot
, where instead, with a single one, it was going in protection in few seconds, with the same resistor (i suspect i burned the single one, at the end, anyway was a second-hand one, so no problems
).