One of the bonuses of building a jumper account and building it up before you jump is so that you can be more powerful than the new players in the new kingdom you jump to. The most frequent advice given to players building a jumper account is to win as many events as possible while you are in the low SH5 bracket. This typically is somewhat hard and time consuming but we have trick that will make it super easy.
The trick is specifically for events that ask you to "increase your building power" or "increase your power" in general. Also, you must stay at SH5 level or lower so that you can still jump to a new opening kingdom with your beginner teleport.
With any building just:
Demolish > Rebuild > Demolish > Rebuild > Demolish > Rebuild > Demolish > Rebuild
over and over again until you reach the amount of points needed to complete the event requirements
Just make sure you have the resources required to demolish and rebuild the building. Most likely you can farm these resources easily, or get them from your main account.
Learn more about Kingdom Jumping by reading these posts: Kingdom Jump Like A Pimp, Changes To Kingdom Jumping, and How UKc took K80 by Storm.
Enjoy!
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Suggested by Fastpowers