Actually, INSERT..EXEC will try to insert BOTH datasets into the table. If the column counts match and the datatype can be implicitly converted, then you will actually get both.
Otherwise, it will always fail because there is no way to only get one of the resultsets.
The solution to this problem is to extract the functionality that you want from the called procedure and incorporate it into the (formerly) calling procedure. And remind yourself while doing it that "SQL is not like client code: redundant code is more acceptable than redundant data".
In case this was not clear above, let me delineate the facts and options available to anyone in this situation:
1) If the two result sets returned are compatible, then you can get both in the same table with the INSERT and try to remove the ones that you do not want.
2) If the two result sets are incompatible then INSERT..EXEC cannot be made to work.
3) You can copy the code out of the called procedure and re-use it in the caller, and deal with the cost of dual-editing maintenance.
4) You can change the called procedure to work more compatibly with your other procedures.
Thats it. Those are your choices in T-SQL for this situation. There are some additional tricks that you can play with SQLCLR or client code but they will involve going about this a little bit differently.