For all open Riemann surface $\mathcal{N}$ and real number $\theta \in (0, \pi/2)$, we construct a conformal minimal immersion $X = (X_1,X_2,X_3) : \mathcal{N} \to \mathbb{R}^3$ such that $X_3+\tan(\theta)\left|X_1\right| : \mathcal{N} \to \mathbb{R}$ is positive and proper. Furthermore, $X$ can be chosen with an arbitrarily prescribed flux map.

Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in $\mathbb{R}^3$ lying above a negative sublinear graph.