  We prove the existence of a one-parameter family of self-similar solutions
with time-dependent tails for Smoluchowski's coagulation equation, for a class
of rate kernels $K(x,y)$ which are homogeneous of degree $\gamma\in(-\infty,1)$
and satisfy $K(x,1)\sim x^{-a}$ as $x\to 0$, for $a=1-\gamma$. In particular,
for small values of a parameter $\rho>0$ we establish the existence of a
positive self-similar solution with finite mass and asymptotics
$A(t)x^{-(2+\rho)}$ as $x\to\infty$, with $A(t)\sim\rho
t^\frac{\rho}{1-\gamma}$.
