  In this paper we present cosmological constraints on several well-known
$f(R)$ models, but also on a new class of models that are variants of the
Hu-Sawicki one of the form $f(R)=R-\frac{2\Lambda}{1+b\;y(R,\Lambda)}$, that
interpolate between the cosmological constant model and a matter dominated
universe for different values of the parameter $b$, which is usually expected
to be small for viable models and which in practice measures the deviation from
General Relativity. We use the latest growth rate, Cosmic Microwave Background,
Baryon Acoustic Oscillations, Supernovae type Ia and Hubble parameter data to
place stringent constraints on the models and to compare them to the
cosmological constant model but also other viable $f(R)$ models such as the
Starobinsky or the degenerate hypergeometric models. We find that these kinds
of Hu-Sawicki variant parameterizations are in general compatible with the
currently available data and can provide useful toy models to explore the
available functional space of $f(R)$ models, something very useful with the
current and upcoming surveys that will test deviations from General Relativity.
