WikiRhymer has about 6,000 rhyming words and phrases for "happier" starting with...

Nappier
    Pure Rhyme Set #1 of 1 (Add to/Edit Set Members)
    Words that have a pure rhyme on their last syllable only.
    barrier
      carrier
        Charrier
          End Rhyme Set #1 of 576 (Add to/Edit Set Members)
          rapier
            Napier
              End Rhyme Set #2 of 576 (Add to/Edit Set Members)
              flashier
                Brashier
                  End Rhyme Set #3 of 576 (Add to/Edit Set Members)

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                  Words that "almost" rhyme on the vowel-based rhyme sound of the stressed syllable like: be/eat or maybe/shapely. (Click for more info)
                  barrier
                    carrier
                      Charrier
                        Near Rhyme Set #1 of 70 (Add to/Edit Set Members)
                        flashier
                          Brashier
                            Near Rhyme Set #2 of 70 (Add to/Edit Set Members)
                            handier
                              Skandia
                                Near Rhyme Set #3 of 70 (Add to/Edit Set Members)

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                                Rhymes made up of more than one word. For instance, "jealous" and "tell us" or "shaky" and "make me."
                                • happy + blur
                                • happy + fer
                                • happy + her
                                • happy + were
                                • unhappy + cur
                                • unhappy + her
                                • unhappy + were
                                Mosaic Rhyme Set #1 of 1 (Add to/Edit Set Members)