A kind of supercompactness Is there a notion of supercompactness of a cardinal $\kappa$ that implies the following?
For every sequence $\langle \alpha_i : i < \kappa \rangle \subseteq \kappa$ and every $i_0 < \kappa$, there is an elementary embedding $j : V \to M$ such that if $\lambda = j(\langle \alpha_i : i < \kappa \rangle)(\kappa + i_0)$, then $M^\lambda \subseteq M$.
This works with almost-hugeness, but I’m wondering if there is an intermediate notion (which already has a definition, or a simpler one that just stating the above).
 A: Your notion is the same as Shelah-for-supercompact. The part with
$i_0$ can be eliminated by suitable translation with a new function. Indeed, the property is equivalent to handle all $i<\kappa$ at once, and even much more than this, as in statement 4. 
Theorem. The following are equivalent for any cardinal
$\kappa$.


*

*Your definition. For every $f:\kappa\to\kappa$ and every
$i_0<\kappa$, there is an elementary embedding $j:V\to M$ into a
transitive class $M$ with critical point $\kappa$ and
$M^{j(f)(\kappa+i_0)}\subset M$.

*Shelah-for-supercompact. For every $f:\kappa\to\kappa$, there is
$j:V\to M$ with critical point $\kappa$ and
$M^{j(f)(\kappa)}\subset M$.

*Uniform version of your definition. For every
$f:\kappa\to\kappa$, there is $j:V\to M$ with critical point
$\kappa$ such that $M^{j(f)(\kappa+i)}\subset M$ for every
$i<\kappa$.

*Souped-up uniform version. For every $f:\kappa\to\kappa$ and every $h:\kappa\to\kappa$, there is $j:V\to M$ with critical point $\kappa$ and $M^{j(f)(\kappa+i)}\subset M$ for all $i<j(h)(\kappa)$. 
In statement 4, for example, suitable choices of $h$ will enable you to handle all $i<\kappa^+$ or $i<(2^{2^\kappa})^M$ and much more: any bound below $j(\kappa)$ that is definable from $\kappa$ in $M$. 
Proof. ($4\to 3\to 1\to 2$) Immediate. 
($2\to 4$). Fix any $f:\kappa\to\kappa$ and $h:\kappa\to\kappa$. Let
$g(\alpha)=\sup_{i<h(\alpha)}f(\alpha+i)$. So $g:\kappa\to\kappa$. By
$2$ there is $j:V\to M$ with critical point $\kappa$ and
$M^{j(g)(\kappa)}\subset M$. Since $j(g)(\kappa)\geq
j(f)(\kappa+i)$ for every $i<j(h)(\kappa)$, this implies also
$M^{j(f)(\kappa+i)}\subset M$ for all $i<j(h)(\kappa)$, verifying (4).
$\Box$
You can find more about large cardinals in this vicinity in the paper of my student Norman Perlmutter:
Perlmutter, Norman Lewis, The large cardinals between supercompact and almost-huge, Arch. Math. Logic 54, No. 3-4, 257-289 (2015). ZBL1371.03071. (arXiv:1307.7387)
Here is a diagram of the large cardinal hierarchy in the vicinity from that paper:

