Other than having the Haste spell cast on you and using the action provided to use something that requires a bonus action are there any other ways to gain additional bonus actions in 5e?
No, there are no ways to gain an additional bonus action. Note that I didn't say no others - the Haste spell doesn't give you one either.
Until the spell ends, the target [...] gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
This isn't an additional bonus action, it's an additional action.
It's hard to prove a negative, but as of 22/01/2018, there are no officially published methods (or Unearthed Arcana) of gaining additional bonus actions.
Only by a few high-level abilities which give you entire new turns during a round.
9th level spell Time Stop:
You briefly stop the flow of time for everyone but yourself. No time passes for other creatures, while you take 1d4 + 1 turns in a row, during which you can use actions and move as normal.
... which includes bonus actions.
The Rogue class Thief archetype has a 17th level ability called Thief's Reflexes, which also similarly gives an additional bonus action as part of giving you a whole additional turn:
You can take two turns during the first round of any combat. You take your first turn at your normal initiative and your second turn at your initiative minus 10.
Short of these, no. Limiting bonus actions to just the one appears to be a fundamental design point. Time Stop takes a highest-tier spell, and even then, the extra actions are quite limited (the spell ends if your actions affect other creatures!). And the Thief ability requires a high character level (the same level that casters are getting 9th level spells, in fact.) This strongly indicates that limiting actions per turn in general is a big concern to 5E.
In addition to the previously mentioned "Time Stop" and "Thief's Reflexes" abilities, "Xanathar's Guide to Everything" has the Samurai subclass for Fighters that get a feature called "Strength Before Death" at level 18- it gives another turn interrupting the current turn as a reaction when the Samurai would drop to 0 hit points. If the Samurai dropped to 0 hit points during their own turn, they would be able to use a Bonus Action on their initial turn and during this new turn before any other creature has a turn.
Additionally, since you've mentioned Haste in your initial question, I think it's worth noting that all Fighters can use "Action Surge" at level 2 for an additional action on their turn, and "Unearthed Arcana: The Faithful" introduced the Seeker Patron for Warlocks which have the "Astral Refuge" ability at level 6 that allows two additional limited actions on their turn; neither of these options would allow for an extra bonus action however.
There is one way (and only one, AFAIK) to get two bonus actions in a round, and that is the 17th level Rogue's power of "Thief's Reflexes". that gives him two full turns in the first round of any combat.