Let $E$ be an elliptic curve over a number field $F$, or a nonconstant elliptic surface (elliptic curve over a function field $F$), and $P$ a point of nonzero canonical height $h$ on $E$. Let $d$ be $\log \vert N(\textrm{disc} (E))\vert$ in the former case, and the discriminant degree of $E$ in the latter. Lang conjectured that there exists a constant $c > 0$ such that $h \ge (c - o(1))D$. Hindry and Silverman proved this under the ABC conjecture for $F$; in particular they proved it unconditionally for an elliptic surface. The constant $c$ obtained by Hindry and Silverman, though explicit, is very small - less than $10^{-10}$ - but is essentially the best that the analysis of Hindry and Silverman can yield. We review this analysis and give two improvements that together improve $c$ to about $1/25000$. We also explain how the same ideas lead to the conjecture that the correct value is $3071/10810800$ (approximately $1/3520$), and compare this with numerical data for explicit $(E,P)$ with low $h$ in the cases $F = \mathbb{Q}$ and $F=\mathbb{C}(t)$.