We generalize a Furstenberg-type result of Orponen-Shmerkin to higher dimensions, leading to an $ε$-improvement in Kaufman's projection theorem for hyperplanes and an unconditional discretized radial projection theorem in the spirit of Orponen-Shmerkin-Wang. Our proof relies on a new incidence estimate for $δ$-tubes and a quasi-product set of $δ$-balls in $\mathbb{R}^d$.