We explore the phase diagram of the O($n$) loop model on the square lattice
in the $(x,n)$ plane, where $x$ is the weight of a lattice edge covered by a
loop. These results are based on transfer-matrix calculations and finite-size
scaling. We express the correlation length associated with the staggered loop
density in the transfer-matrix eigenvalues. The finite-size data for this
correlation length, combined with the scaling formula, reveal the location of
critical lines in the diagram. For $n>>2$ we find Ising-like phase transitions
associated with the onset of a checkerboard-like ordering of the elementary
loops, i.e., the smallest possible loops, with the size of an elementary face,
which cover precisely one half of the faces of the square lattice at the
maximum loop density. In this respect, the ordered state resembles that of the
hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of
$n$ represents a softening of its particle-particle potentials. We also
determine critical points in the range $-2\leq n\leq 2$. It is found that the
topology of the phase diagram depends on the set of allowed vertices of the
loop model. Depending on the choice of this set, the $n>2$ transition may
continue into the dense phase of the $n \leq 2$ loop model, or continue as a
line of $n \leq 2$ O($n$) multicritical points