Expressing 2N point DFT in terms of N point DFT
Consider a sequence $x[n]$ of length $N$ whose $N$-point DFT is $X[k]$. Then let $X_2[k]$ be the $2N$-point DFT of $x[n]$. 

As you have stated, the even indexed samples of $X_2[k]$ will be easily shown to be: $$X_2[k] = X[k/2] ~~~,~~~k = 2m, m=0,1,...,N-1$$

Then the odd indexed samples of $X_2[k]$ will be given by the $N$-point DFT of the signal  $x[n] e^{-j \frac{\pi}{N} n } $, $n=0,1,...,N-1$. 

$$X_{2}[k] = \sum_{n=0}^{N-1} x[n] e^{-j \frac{\pi}{N} n} e^{-j \frac{2\pi}{N} k n}  = X[k + 0.5]  ~~~~ ,~~~ k =2m+1, m=0,1,...,N-1 $$