  Pure {\it compact} $U(1)$ lattice gauge theory exhibits a phase transition at
gauge coupling $g \sim {\cal{O}}(1)$ separating a familiar weak coupling
Coulomb phase, having free massless photons, from a strong coupling phase.
However, the phase transition was found to be of first order, ruling out any
nontrivial theory resulting from a continuum limit from the strong coupling
side. In this work, a compact $U(1)$ lattice gauge theory is studied with
addition of a dimension-two mass counterterm and a higher derivative (HD) term
that ensures a unique vacuum and produces a covariant gauge-fixing term in the
naive continuum limit. For a reasonably large coefficient of the HD term, now
there exists a continuous transition from a regular ordered phase to a
spatially modulated ordered phase. For weak gauge couplings, a continuum limit
from the regular ordered phase results in a familiar theory consisting of free
massless photons. For strong gauge couplings with $g\ge {\cal{O}}(1)$, this
transition changes from first order to continuous as the coefficient of the HD
term is increased, resulting in tricritical points which appear to be a
candidate in this theory for a possible nontrivial continuum limit.
