  Let $M$ be a closed orientable irreducible $3$-manifold such that $\pi_1(M)$ is left orderable.   (a) Let $M_0 = M - Int(B^{3})$, where $B^{3}$ is a compact $3$-ball in $M$. We have a process to produce a co-orientable Reebless foliation $\mathcal{F}$ in $M_0$ such that: (1) $\mathcal{F}$ has a transverse $(\pi_1(M),\mathbb{R})$ structure, (2) there exists a simple closed curve in $M$ that is co-orientably transverse to $\mathcal{F}$ and intersects every leaf of $\mathcal{F}$. More specifically, given a pair $(<,\Gamma)$ composed of a left-invariant order "$<$" of $\pi_1(M)$ and a fundamental domain $\Gamma$ of $M$ in its universal cover with certain property (which always exists), we can produce a resulting foliation in $M - Int(B^{3})$ as above, and we can test if it can extend to a taut foliation of $M$.   (b) Suppose further that $M$ is either atoroidal or a rational homology $3$-sphere. If $M$ admits an $\mathbb{R}$-covered foliation $\mathcal{F}_0$, then there is a resulting foliation $\mathcal{F}$ of our process in $M - Int(B^{3})$ such that: $\mathcal{F}$ can extend to an $\mathbb{R}$-covered foliation $\mathcal{F}_{extend}$ of $M$, and $\mathcal{F}_0$ can be recovered from doing a collapsing operation on $\mathcal{F}_{extend}$. Here, by a collapsing operation on $\mathcal{F}_{extend}$, we mean the following process: (1) choosing an embedded product space $S \times I$ in $M$ for some (possibly non-compact) surface $S$ such that $S \times \{0\}, S \times \{1\}$ are leaves of $\mathcal{F}_{extend}$ (notice that $\mathcal{F}_{extend} \mid_{S \times I}$ may not be a product bundle), (2) replacing $\mathcal{F}_{extend} \mid_{S \times I}$ by a single leaf $S$.   (c) We conjecture that there always exists a resulting foliation of our process in $M - Int(B^{3})$ which can extend to a taut foliation in $M$. 
Left orderability, foliations, and transverse $(\pi_1,\mathbb{R})$   structures for $3$-manifolds with sphere boundary