  We have analysed the recently extended series for the number of self-avoiding
walks (SAWs) $C_L(1)$ that cross an $L \times L$ square between diagonally
opposed corners. The number of such walks is known to grow as
$\lambda_S^{L^2}.$ We have made more precise the estimate of $\lambda_S,$ based
on additional series coefficients provided by several authors, and refined
analysis techniques. We estimate that $\lambda_S = 1.7445498 \pm 0.0000012.$ We
have also studied the subdominant behaviour, and conjecture that $$ C_L(1) \sim
\lambda_S^{L^2+bL+c}\cdot L^g,$$ where $b=-0.04354 \pm 0.0001,$ $c=0.5624 \pm
0.0005,$ and $g=0.000 \pm 0.005.$
  We implemented a very efficient algorithm for enumerating paths on the square
and hexagonal lattices making use of a minimal perfect hash function and
in-place memory updating of the arrays for the counts of the number of paths.
  Using this algorithm we extended and then analysed series for SAWs spanning
the square lattice and self-avoiding polygons (SAPs) crossing the square
lattice. These are known to also grow as $\lambda_S^{L^2}.$ The sub-dominant
term $\lambda^b$ is found to be the same as for SAWs crossing the square, while
the exponent $g = 1.75\pm 0.01$ for spanning SAWs and $g = -0.500 \pm 0.005$
for SAPs.
  We have also studied the analogous problems on the hexagonal lattice, and
generated series for a number of geometries. In particular, we study SAWs and
SAPs crossing rhomboidal, triangular and square domains on the hexagonal
lattice, as well as SAWs spanning a rhombus. We estimate that the analogous
growth constant $\lambda_H=1.38724951 \pm 0.00000005,$ so an even more precise
estimate than found for the square lattice. We also give estimates of the
sub-dominant terms.
