Sometimes the inflate method is best, sometimes not.  Probably best when N is slightly less than a power of 2.

Sometimes Paul's modification is best:  call it overinflation, to go higher than the next power of 2.

In the case of N=9, it is better still to factor 9 = 3*3 and apply inflate(3) twice.

inflate(9):  4 * 16/9 = 7.111

overinflate(9):  6 * 64/63 = 6.1

factor to 3*3:  2 * 4/3   +   2 * 4/3 = 5.333

I have not found any examples where the "trick" I discussed in my second post actually helps.

For several small even numbers I checked, it is better to do individual flips to divide the group in half each time, then when an odd number is reached, apply the inflate method.  The inflate method is not always optimal, but dividing by two (I think) always helps.

For N=6 for example, do one flip to split the group in half, then apply inflate(3):  

  1 + 2 * (4/3) = 11/3 = 3 2/3 which matches what Paul found.

Right now, I'm thinking it is always better to divide by 2 and sometimes better to divide by other factors, I'm guessing that whatever it takes to get close to 2^k - 1  (inflate, vs overinflate, vs divide by a factor) is the winning strategy.