In this paper we investigate time-dependent Kantowski-Sachs spherically symmetric teleparallel $F(T)$ gravity with a scalar field source. We begin by setting the exact field equations to be solved and solve conservation laws for possible scalar field potential $V\left(\phi\right)$ solutions. Then we proceed to find new non-trivial teleparallel $F(T)$ solutions by using power-law and exponential ansatz for each potential cases arising from conservation laws such as linear, quadratic, logarithmic to name a few. We find a general formula allowing to compute all possible new teleparallel $F(T)$ solutions applicable for any scalar field potential and ansatz. Then we apply this formula and find a large number of exact and approximate new teleparallel $F(T)$ solutions for several types of cases. Some new $F(T)$ solution classes may be relevant for future cosmological applications, especially concerning the dark matter, the dark energy quintessence, phantom energy leading to the Big Rip event and quintom models of physical processes.