Assume that $M$ is a compact Riemannian manifold. The unit tangent bundle of $M$ is denoted by $SM$ with the natural projection $p:SM \to M$.
A connection on $SM$ is a distribution for $SM$ with trivial intersection with the vertical distribution $\ker Dp$
Is there always a connection $T $ on $SM$ with the following property?:
For every $(x,v) \in SM$ we have $Dp(T_{(x,v)}) \perp v$
If the answer is yes, can we always chose such type of connection with dimension $n-1$?
Since parallel translates of unit vectors are unit vectors, the Riemannian connection on $TM$ restricts to a natural (Ehresmann; i.e. $n$-dimensional in your parlance) connection $H$ on $SM$, giving us a splitting $TSM = H \oplus \ker Dp$. In particular this implies $Dp|_H : H \to p^* TM$ is a bundle isomorphism over $\mathrm{id}_{SM}$, so we can smoothly identify $TSM = p^* TM \oplus \ker Dp$.
After making this identification, there is an obvious choice for your $(n-1)-$dimensional "connection": we can just let $T = W \oplus \{0\}$ where $W$ is the subbundle of $p^* TM$ defined by $W_{(x,v)} = v^\perp = \{w \in T_x M: \langle v,w \rangle = 0 \}.$