We study the existence and scattering of global small amplitude solutions to modified improved Boussinesq equations in one dimension with nonlinear term f(u) behaving as a power up as u→0. Solutions are considered in Hs space for all s>1/2. According to the value of s, the power nonlinearity exponent p is determined. Liu (Liu 1996 Indiana Univ. Math. J. 45, 797-816) obtained the minimum value of p greater than 8 at s=3/2 for sufficiently small Cauchy data. In this paper, we prove that p can be reduced to be greater than 9/2 at s>17/10 and the corresponding solution u has the time decay, such as ‖u(t)‖L∞=O(t-2/5) as t→∞. We also prove non-existence of non-trivial asymptotically free solutions for 1<p≤2 under vanishing condition near zero frequency on asymptotic states.