There are numerous large cardinal notions that are upward reflecting in the sense you have described. On the one hand, some of the very largest notions have much stronger versions of the upward reflecting property, since indeed the large cardinal property $p$ itself reflects upward, in the sense that every instance of a $p$ cardinal has largest instances of $p$ cardinals.

For example, if $\kappa$ is a rank-into-rank cardinal, meaning that it is the critical point $\kappa$ of an elementary embedding $j:V_\lambda\to V_\lambda$, then $j(\kappa)$ is also a rank-into-rank cardinal, witnessed by the application $j(j)$, the union $\bigcup_{\alpha<\lambda}j(j\upharpoonright V_\alpha)$, which is an elementary embedding $V_\lambda\to V_\lambda$ with critical point $j(\kappa)$.

A similar argument works with $j:V_{\lambda+1}\to V_{\lambda+1}$ and $j:L(V_\lambda)\to L(V_\lambda)$, which are amongst the very strongest large cardinal properties.

If $\kappa$ is Reinhardt (considered in GB without AC), witnessed by $j:V\to V$, then $j(\kappa)$ is similarly Reinhardt, witnessed by $j(j)$.

A bit lower, we have further examples with the huge cardinals, which are the critical points of embeddings $j:V\to M$ where we have $M^{j(\kappa)}\subset M$. This implies $V_{j(\kappa)}\subset M$ and since $j(\kappa)$ is a large cardinal in $M$, it is a limit of many inaccessible and much larger large cardinals in $M$ which must therefore have these properties also in $V$. Similar reasoning applies to the almost huge and superhuge cardinals, as well as to the superstrong cardinals.

In particular, this reasoning applies to Shelah cardinals. A cardinal $\kappa$ is *Shelah* if for every $f:\kappa\to\kappa$ there is $j:V\to M$ with critical point $\kappa$ and $V_{j(f)(\kappa)}\subset M$. If you use the function $f(\gamma)=$next inaccessible after $\gamma$ plus $1$, then the witnessing $j:V\to M$ will have $V_{j(f)(\kappa)}=V_{\delta+1}\subset M$, where $\delta$ is the next inaccessible cardinal after $\kappa$ in $M$. In particular, this implies that $\delta$ really is inaccessible in $V$, and so Shelah cardinals are upward reflecting in your sense.

Meanwhile, Woodin cardinals are not upwardly reflecting, and indeed, there are numerous large cardinals that do not necessarily exhibit your upward reflecting property, including measurable cardinals, strong cardinals, strongly compact cardinals, supercompact cardinals, and many others. The reason is that if $\kappa$ exhibits any of these large cardinal properties, and $\lambda\gt\kappa$ is the next inaccessible cardinal above $\kappa$, then $\kappa$ will continue to exhibit the large cardinal inside $V_\lambda$, where there will be no inaccessible cardinals above $\kappa$. (Note, we may work in a model of GCH, where the distinction between strongly inaccessible and weakly inaccessible evaporates.) Basically, any large cardinal property that is witnessed sufficiently locally, so that it remains true whenever one cuts off the universe at an inaccessible level, will fail to be upwardly reflecting in your sense.

Lastly, let me mention that one shouldn't think that it is only the very large large cardinals that are upward reflecting. Consider the uplifting cardinals, which are rather weak in consistency strength, below Mahlo cardinals. The uplifting cardinals and even the pseudo uplifting cardinals are upward reflecting, since they are inaccessible cardinals $\kappa$ for which there is an elementary extension $V_\kappa\prec V_\lambda$, and this implies that $\lambda$ must be a limit of inaccessible cardinals above $\kappa$.