If $m$ is even, $d_m$ is multiplication by $2$:

For every unit sphere bundle of an $n$-dimendional vector bundle over a space $X$, $d_m$ sends $[S^{n-1}] \otimes 1$ to the euler class $e(x) \in H^n(X)$. In your case, we know that $$\langle e, [S^m]\rangle = \chi(S^m) = 1+(-1)^n$$ whcih is $2$ if $m$ is even. You can read off all the cohomology of $\tau(S^n)$ from that.

If $m$ is even, $d_m$ is multiplication by $2$:

For every unit sphere bundle of an $m$-dimendional vector bundle over a space $X$, $d_m$ sends $[S^{m-1}] \otimes 1$ to the euler class $e(x) \in H^m(X)$. In your case, we know that $$\langle e, [S^m]\rangle = \chi(S^m) = 1+(-1)^m$$ which is $2$ if $m$ is even. You can read off all the cohomology of $\tau(S^m)$ from that.