  We use a new approach to color evaporation model (CEM) for quarkonium
production. The production of $c\bar c$ pairs is performed within
$k_T$-factorization approach using different unintegrated gluon distribution
functions (UGDF) from the literature. We include all recent improvements to
color evaporation model. We get poor description of the large transverse
momentum distributions of $J/\psi$ with the JH-2013 CCFM-based UGDF. Here
explicit inclusion of $2 \to 3$ processes considerably improves the situation.
Similar effects are discussed in the context of the KMR UGDF.
