The Excel does not have a default function to add labels both inside and outside, however, with a few of tips, you can make your chart perfectly with labels in and out. The logic behind is to make two exactly same pie charts but with different labels.
For example, suppose we have the data below and we are going to make a chart with percentage labels inside and Names outside.
Step 1: To create a regular pie chart (see here). Select first two columns of data, then in the Insert Tab from Ribbon, click Pie Chart. A basic pie chart will be created;
Step 2: Delete Legend at the bottom (based on your setting, legend may appear in other position);
Step 3: Add Data Labels to the pie chart: right click on the pie, then click "Add Data Label";
The data labels were added to the pie chart.
Step 4: "Category Name" and Position: Right click on any data label, and select "Format Data Labels", in the dialog window, check "Category Name", "Show Leader Lines" and then check "Outside End" in the Label Position section.
Now, Pie chart should look like this:
Step 5: Rotate Pie Chart: To rotate the pie chart, right click on the pie, select "Format Data Series", adjust the "Angle of first slice" to any degree you need (e.g., 64 degree, this number will be used later);
Step 6: To have the Leader Lines, drag each of the category names a bit far from the pie;
Step 7: To create another chart exactly the same on top of this one. Copy the first two columns data and paste on the pie chart (right click on anywhere in the plot area but not directly on the pies);
This step will generate a second pie chart but you can not see any changes.
Step 8: Move up the newly pasted chart up;
1. Right click on the pie chart, click "Select Data";
2. In the "Select Data Source" window, click move down button, the newly copied chart will move up; at the same time, you will notice all labels disappeared;
3. You should have a pie chart same as below;
Step 9: Add data labels in the NEW pie chart;
1. Right click on the pie chart, click "Add Data Labels";
2. Right click on the data label, click "Format Data Labels" in the dialog box;
3. In the "Format Data Labels" window, select "value", "Show Leader Lines", and then "Inside End" in the Label Position section;
Step 10: Set second chart as Secondary Axis:
1. Right click on the pie chart, then click "Format Data Series";
2. In the "Format Data Series" window, select "Secondary Axis";
3. The chart should like the one below;
Step 11: Rotate to match the first pie chart (In Step 5, the first pie chart rotated 64 degrees);
1. Right click on the pie chart, then select "Format Data Series";
2. Change the "Angle of first slice" to 64 degrees (or the same degree you set up in step 5);
3. The pie now should look like the one below with percentage inside and Category Names outside with Leader Lines.
4. You may need to work on other elements to make the chart nicer, such as label font, bold, size, position, chart title, legends, and borders, etc.