  We prove a class of modified paraboloid restriction estimates with a loss of
angular derivatives for the full set of paraboloid restriction conjecture
indices. This result generalizes the paraboloid restriction estimate in radial
case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127-1168], as well as the result
from [Miao et al. Proc. AMS 140(2012), 2091-2102]. As an application, we show a
local smoothing estimate for a solution of the linear Schr\"odinger equation
under the assumption that the initial datum has additional angular regularity.
