-Some people have the custom not to wear new clothing during Sefira. This custom, too, has no real source in Halacha, and most likely developed from a misapplied comparison to the prohibition of wearing new clothing during the 3 weeks. (See Shu”t Yabia Omer Vol. 3 Siman 26)
If one does indeed have this custom in his/her family for many generations, they should abide by it. However, if this isn’t the custom in your family, new clothing may be worn. (See Sefer Bain Pesach L’Shavuos Chapter 16 footnote 1 near the end what he quotes from the Yosef Ometz Siman 845 and 848 to explain why certain communities had this custom, and why it doesn’t apply to everyone)-Even those whose custom it is not to wear new clothing during Sefira may purchase new clothing in Sefira.
Additionally those who avoid wearing new clothing only avoid significant articles of clothing (such as a suit or dress and the like) but not other articles (such as socks, undergarments, shirts etc.).
Also, dressing children in new clothing, is treated more leniently even by those who themselves avoid new clothing. (See Shu”t Vayevarech Dovid Siman 59. See also Leket Yosher page 98 regarding shoes.)