The double covering $$ \text{Spin}(n)\to SO(n) $$ is the universal covering for $n\geq 3$ because $\pi_1(SO(n)) = \mathbb{Z}_2$ in this case. Analogously, $$ \text{Spin}(1,n)\to SO^+(1,n) $$ is the universal covering for $n\geq 3$ because $$ SO(1,3)\cong SO(1)\times SO(n)\times\mathbb{R}^{n} $$ has the same fundamental group of $SO(n)$.
Are these the only signatures when the $\text{Spin}(s,t)\to SO^+(s,t)$ is the universal covering?