There are other situations that necessitate Halachic hand washing. Several are due to Ruach Ra’ah, which rests on ones hands in certain situations much as after sleeping, although the washing need not be with the same Halachic urgency, however one should wash as soon as possible.
Entering a primitive bathroom and entering a bathhouse, even without using the facilities; Cutting nails and cutting hair; and touching parts of the body that are usually covered require washing until the wrist but only once on each hand (The Shelah writes that in all these cases one should wash three times).
Going to a cemetery, touching a dead body and having marital relations require three times according to Shaarei Tesuva because they have a higher level of Ruach Ra’ah. There is a minority opinion that includes going to the bathroom in this category.
Touching used leather shoes, touching sweaty areas or clothing, scratching your scalp, touching ear or nose waste (according to some), and changing a diaper all require hand washing due to cleanliness. This is different than Ruach Ra’ah in the respect that not the whole hand need be washed. According to some opinions marital relations and touching usually covered parts of the body are also included in this category.
A little more in depth:
Cutting nails – however they’re cut, and even if they are cut by someone else (a manicurist) although the person cutting does not have to wash their hands. This applies to both finger and toe nails. Nails that are bitten as a way of cutting them also require washing. For more on fingernails look here.
Cutting hair – Both the barber and the person getting a haircut (Barbie?) must wash their hands. Rav Shlomo Zalman Auerbach has two interesting rulings here. If one takes a snip of a 3 year old boys hair at his upsherin (as is customary) he need not wash his hands, and after cutting the beard (or shaving) and payos one need not wash his hands.
Bathroom – According to most opinions modern bathrooms do not necessitate hand washing if one entered them and did not relieve themselves. [a port a potty would be different].
Parts of the body that are usually covered – in all places will include above the knees, the entire torso and back, and above the elbows. Elsewhere would depend on whether they are usually covered. It is questionable if this is because of Ruach Ra’ah or cleanliness and therefore one should wash their whole hand.
Funerals and cemeteries – if one was in the same building as, or came within 4 amos of, a decedent he must wash his hands. The minhag as recorded by the Rema is to wash ones hands before reentering a building. Rav Moshe Feinstein said that this only applies to a residence, not a public building.
Marital Relations – Since this is required for both Ruach Ra’ah and cleanliness purposes one should wash their whole hand. The Mishna Berura quotes an opinion that requires washing three times.