While I don't specialize in band management or promotion, I have some second-hand advice as my boyfriend is a signed musician to a major label.
From my understanding of your question, it sounds like they have some sort of following already - from here, I'd identify within the US or Europe where you strongest following is - then target that area first for a more local-focused tour. For example, an LA-focused tour, New England or Midwest tour. This will help keep costs low while creating more fans that can act as band promoters.
If there is no following at the moment, you should consider shopping the band and their cover for press coverage to some major music blogs such as pitchfork, ear milk, indiecurrent, etc. Here is a great list of music blogs: http://www.digitalmusicnews.com/permalink/2014/03/26/20blogs. I also know that Perez Hilton often writes about covers that he knows will resonate with his readers.
You could also identify music tastemakers for their style on Spotify and build a relationship with them to see if they're interested in including the band in some of their upcoming/new playlists. Many of these folks like being the "first to" knowledge and up-and-coming bands, so this can be a great and sort of easy go-to.
Find some of the influencers who are a little lower down on the influence-scale (think a 6 or 7 instead of 10's like Pitchfork) and see if they're interested in interviewing the band via Skype, in person, or would provide coverage for "first access" to new releases. Great examples of these blogs are are "We Found New Music" or "Obscure Sound."
Lastly, you could always try to have them signed to a booking agency that best represents their style. Some examples are Windish (https://twitter.com/windishagency) which is large and known for more indie bands. A some new ones like Boston Music Scene (https://twitter.com/bostonmusicscne).
I hope this helps! Promoting music now-a-days is a totally different beast from what it was several years ago, but there are more channels/mediums now, which create more opportunities for them to get heard. Good luck!