EDITED TO REFLECT @LUCIA'S COMMENTS: After digging through a bunch of references, I sorted out the answer I was looking for.  For all $T > 0$ except at the points of discontinuity of $N(2\pi T)$, one has
$$N(2 \pi T) =1+ \frac{1}{\pi} \theta(2\pi T) + \frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right)+2R(T),$$
where $\operatorname{Arg}$ is the principal branch of the argument, where $\theta(T)$ is the Riemann-Siegel theta function, and where $R(T)$ is an integer that is $O(\log T)$.
One has the asymptotic relation
\begin{align*}
N(2 \pi T) & =T \log T-T+\frac{7}{8}+\frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right)+2R(T)+O\left(\frac{1}{T}\right)  \\
& =   \int_1^T \log t \, dt -\frac{1}{8}+\frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right)+2R(T)+O\left(\frac{1}{T}\right) 
\end{align*}
and the asymptotic expansion
\begin{align*}
N(2 \pi T)  - \frac{1}{\pi}\operatorname{Arg}\zeta\left(\frac{1}{2}+2\pi i T\right)   \sim T \log T - T+ 2R(T) + \frac{7}{8}+\frac{1}{96 \pi^2 T}+ \frac{7}{11340\pi^4 T^3}+ \frac{31}{161280 \pi^6 T^5}+\cdots
\end{align*}
as $T \to \infty$, where the numerators and (1/2)denominators are as in OEIS Sequences A036282 and A114721, respectively.  Here is a plot of the function $N(2\pi T)$ and its smooth approximation  $1+\frac{1}{\pi}\theta(2\pi T)$. 

What is the infimum of all $T$ that is not $\frac{1}{2\pi}$ of an imaginary part of a zero of $\zeta(s)$ such that $R(T)\neq 0$? 

According to p. 98 of Davenport's book, you can get rid of the $R(T)$ integer term by replacing the principal value of $\arg \zeta(1/2+2 \pi i T)$ with the variation of $\arg \zeta(s)$ from $s = +\infty+2\pi iT$ to $1/2+2 \pi iT$ starting with value $0$, as long as  $T$ is not $\frac{1}{2\pi}$ times an ordinate of a zero of $\zeta(s)$.
 [![enter image description here][1]][1]


  [1]: https://i.sstatic.net/lf6z8.png