compressing rarely used space
I think it can be compressed to linear space.

I assume the machine $M$ has one tape. Initially the first $n$ cells contain the input, which is followed by $2^O(n)$ zeroes. $M$ is oblivious and always makes $k$ passes through the tape (from left end to right and back), and then accepts or rejects. Let $p_i$ denote the state the machine enters after reading the $n$-th cell on $i$-th pass during the left-right phase, and let $q_i$ denote the state the machine enters after reading the $n+1$-th cell on $i$-th pass during the right-left phase. Note that $q_1$ is a function of $p_1$ alone, and generally $q_i$ depends only on the sequence $p_1, p_2, \dots, p_i$ (this sequence has length at most $k$, which is fixed). Therefore we can always compute the state $q_i$ without actually continuing the pass through the extra scratch space - the transitions can be hardcoded into the description of the machine.