This answer is a complement to Joel David Hamkin's very nice answer.

By recent work of J. Bagaria and M. Magidor, there is a large cardinal notion between measurable and strongly compact that answers your question (as formulated by Hamkins). This notion is $\omega_1$-strongly compact cardinals.

Definition A cardinal $\kappa$ is $\delta$-strongly compact, if every $\kappa$-complete filter on a set $I$ can be extended to an $\delta$-complete ultrafilter on $I$. We focus on the case where $\delta=\omega_1$.

Some consequences of the definition:

1. Every strongly compact cardinal is $\omega_1$-strongly compact.
2. If $\kappa$ is $\omega_1$-strongly compact, then the same is true for every $\lambda\ge\kappa$. So, the interest is in the first $\omega_1$-strongly compact.
3. If $\kappa$ is $\omega_1$-strongly compact and $\mu$ is the first measurable, then $\kappa$ is $\mu$-strongly compact. So, the first measurable can not be above the first $\omega_1$-strongly compact.

It follows that if first measurable=first strongly compact, then first measurable=first $\omega_1$-strongly compact=first strongly compact. This complements the three theorems from Hamkin's answer.

To your question now, Bagaria and Magidor proved the following.

Theorem The following are equivalent:

• $\kappa$ is a strong compactness cardinal for $L_{\omega_1,\omega}$
• $\kappa$ is a strong compactness cardinal for $L_{\omega_1,\omega_1}$
• $\kappa$ is $\omega_1$-strongly compact
• For every set $I$ there is an $\omega_1$-complete fine measure of $P_\kappa(I)$.

They actually prove more since they provide a list of 5-6 equivalent formulations of $\omega_1$-strong compactness. See [these][1] slides from one of Magidor's talks.

In the second reference below, it is proved using Radin forcing that consistently the first $\omega_1$-strongly compact is singular (of measurable cofinality) and therefore, strictly between the first measurable and the first strongly compact. This justifies the claim that (consistently) neither a measurable nor a strongly compact cardinal can answer your question.

References:

Bagaria, Joan; Magidor, Menachem, On $\omega_1$-strongly compact cardinals, J. Symb. Log. 79, No. 1, 266-278 (2014). ZBL1337.03076.

and

Bagaria, Joan; Magidor, Menachem, Group radicals and strongly compact cardinals, Trans. Am. Math. Soc. 366, No. 4, 1857-1877 (2014). ZBL1349.03055. [1]: http://www.crm.cat/en/Activities/Curs_2016-2017/Documents/Tutorial%20lecture%203.pdf

This answer is a complement to Joel David Hamkin's very nice answer.

By recent work of J. Bagaria and M. Magidor, there is a large cardinal notion between measurable and strongly compact that answers your question (as formulated by Hamkins). This notion is $\omega_1$-strongly compact cardinals.

Definition A cardinal $\kappa$ is $\delta$-strongly compact, if every $\kappa$-complete filter on a set $I$ can be extended to an $\delta$-complete ultrafilter on $I$. We focus on the case where $\delta=\omega_1$.

Some consequences of the definition:

1. Every strongly compact cardinal is $\omega_1$-strongly compact.
2. If $\kappa$ is $\omega_1$-strongly compact, then the same is true for every $\lambda\ge\kappa$. So, the interest is in the first $\omega_1$-strongly compact.
3. If $\kappa$ is $\omega_1$-strongly compact and $\mu$ is the first measurable, then $\kappa$ is $\mu$-strongly compact. So, the first measurable can not be above the first $\omega_1$-strongly compact.

It follows that if first measurable=first strongly compact, then first measurable=first $\omega_1$-strongly compact=first strongly compact. This complements the three theorems from Hamkin's answer.

To your question now, Bagaria and Magidor proved the following.

Theorem The following are equivalent:

• $\kappa$ is a strong compactness cardinal for $L_{\omega_1,\omega}$
• $\kappa$ is a strong compactness cardinal for $L_{\omega_1,\omega_1}$
• $\kappa$ is $\omega_1$-strongly compact
• For every set $I$ there is an $\omega_1$-complete fine measure of $P_\kappa(I)$.

They actually prove more since they provide a list of 5-6 equivalent formulations of $\omega_1$-strong compactness. See these slides from one of Magidor's talks.

In the second reference below, it is proved using Radin forcing that consistently the first $\omega_1$-strongly compact is singular (of measurable cofinality) and therefore, strictly between the first measurable and the first strongly compact. This justifies the claim that (consistently) neither a measurable nor a strongly compact cardinal can answer your question.

References:

Bagaria, Joan; Magidor, Menachem, On $\omega_1$-strongly compact cardinals, J. Symb. Log. 79, No. 1, 266-278 (2014). ZBL1337.03076.

and

Bagaria, Joan; Magidor, Menachem, Group radicals and strongly compact cardinals, Trans. Am. Math. Soc. 366, No. 4, 1857-1877 (2014). ZBL1349.03055.