The boundedness of the small Hankel operator $h_f^\nu(g)=P_\nu(f\bar{ g})$, induced by an analytic symbol $f$ and the Bergman projection $P_\nu$ associated to $\nu$, acting from the weighted Bergman space $A^p_\om$ to $A^q_\nu$ is characterized on the full range $0<p,q<\infty$ when $\omega,\nu$ belong to the class $\mathcal{ D}$ of radial weights admitting certain two-sided doubling conditions. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization $A_{ \eta}^{ q}=A_{ \omega}^{ p_{ 1}}\odot A_{ \nu}^{ p_{ 2}}$, where $1<q,p_{ 1},p_{ 2}<\infty$ such that $q^{ -1}=p_{ 1}^{ -1}+p_{ 2}^{ -1}$ and $\widetilde{ \eta}^{ \frac{ 1}{ q}}\asymp\widetilde{ \omega}^{ \frac{ 1}{ p_{ 1}}}\widetilde{ \nu}^{ \frac{ 1}{ p_{ 2}}}$. Here $\widetilde{ \tau}(r)=\int_r^1\tau(t)\,dt/(1-t)$ for all $0\le r<1$.
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