How do I get a double-entry Schengen visa?

Yes, assuming that you will be leaving the Schengen Area in the mean time you should add this information with the event details as part of your itinerary.

This combination would be considered a valid reason for the issuing of a double entry visa.

Page 20 (22 of PDF) of

• Handbook for the processing of visa applications and the modifictaion of issued visas (2019-05-14)

1.2 Application for a uniform visa for multiple entries
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Recommended best practice: In case an applicant is to travel to several Member States on different trips within a short timeframe, consulates should apply a flexible approach and consider this as one trip for the application of Article 5(1) of the Visa Code to avoid unnecessary burden and costs for the visa applicant.

Pages 75-77 (76-78 of PDF)

8. TYPES OF VISA TO BE ISSUED
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8.4 Number of entries
A uniform visa may be issued for one, two or multiple entries. The applicant should indicate in the application form how many entries he wishes, but it is the consulate examining the application who decide on the number of entries to grant. When a multiple-entry visa is issued, the period of validity of the travel document should be respected (see point 8.1.1 above).
one entry
If the applicant's purpose of travel is one particular event defined in time, only one entry should be allowed!
two entries
If the applicant's purpose of travel is one particular event defined in time, but during that visit he wishes to visit the United Kingdom or Ireland, he should be issued a two-entry visa.
multiple entries
When a multiple entry visa is issued with a period of validity between 180 days (6 months) and 5 years, the duration of authorised stay is always 90 days (per 180 day-period). This also applies in cases where the visa is issued for the purpose of transit.

Visas allowing for multiple entries may also be issued with a validity of less than 6 months.

Multiple-entry visa should, in principle, not be issued to first-time applicants.
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Recommended best practice with regard to the harmonised practices for the issuing of multiple entry visas in a given location: With a view to harmonising practices as part of local Schengen cooperation, Member States should discuss the length of validity of multiple entry visas to be issued to persons eligible for such visas and the progressively increasing validity of such visas in accordance with the below guidelines, which may be adapted to local circumstances:

• persons having lawfully used previous Schengen visas issued within the previous year: a multiple-entry visa with a validity of one year;
• persons having lawfully used a multiple entry Schengen visa, in the previous 24 months: a multiple-entry visas with a validity of a minimum of two and a maximum of five years.

Sources:

• Handbook for the processing of visa applications and the modifictaion of issued visas (2019-05-14)

I assume you stay in the Schengen area for your German summer school and your Hungary event.
If so, you should explain the Hungary event when applying for your Germany visa. A single entry visa will cover you for this kind of trip.
If your trip to Hungary is minor compared to the time you spend in Germany, it might even be acceptable not to mention it on applying for your visa, but as it is quite a distance I would tell them.

If on the other hand, you have to leave the Schengen area before or after your Hungary trip, you will need a two entry or multiple entry visa (for three or more entries.) Then again you need to explain the whole set of travels when applying for the German visa, if the visa officials do what is the most expected you will get the visa for the whole trip. But you are not guarantied to get a multiple visa, so if you can organize your trip such that you do not need the Schengen area.

If applying for just one visa for the whole period would get you over 90 days, applying for two or more visa in one go or separately, will not help you, as you will still only get 90 days out of 180 days on a tourist visa. It may mean that you have to apply for a different kind of visa, like a study visa. The only way around that maximum of 90 days is to have a break out of the Schengen area so your maximum time is 90 days or less. Hungary is within Schengen, so going there will not stop your days counter.