The two only linear codes I know, hamming and minitel encode by preserving the message and appending something at the end but I know some codes do not have this property. So I was wondering if the following statement was true:
"For any linear code, you can find a linear code as efficient (detects the same number of errors, corrects the same number of errors and uses as many additional bits) that keeps the original message at the beginning of the encoded message."
Yes, this is known as systematic encoding and it can always be achieved without (essentially) changing the code. The reason for this comes from linear algebra. A linear code is the row space of its generator matrix. The row space of any matrix is preserved, when we do elementary row operations on it. Therefore we can assume that the generator matrix is in a reduced row echelon form. After that step we may need to do some number of column swaps, so that we get an identity block into the leftmost columns. This does change the row space, because it amounts to permuting some components of the vectors. But because the same component permutation will be applied to all the words of the code, its Hamming distance and error-correcting capability are preserved.
In coding theory we say that two binary linear codes are equivalent, if they are gotten from each other by doing the same permutation of components to all the words.