The Legion of Space by Jack Williamson matches parts of your description: invading aliens from Barnard's star; hero is a military guy (Legion of Space); FTL space drive; hero travels to alien homeworld with aliens in pursuit; alien jungle; girl. Also the answer to this old question and this one. It was originally published as a six-part serial in the April, May, June, July, August, and September, 1934 issues of of Astounding Stories, which are available at the Internet Archive (, , , , , ). Here is an extract from Everett F. Bleiler's review in Science-Fiction: The Gernsback Years:
Special Background: Adam Ulnar, commander of the Legion of Space, is conspiring to place his nephew Eric back on the throne. Eric has just returned from an interstellar expedition to Barnard's Runaway Star, where he has encountered the alien civilization of the Medusae. The Medusae, who are able to levitate themselves by manipulating space-time coordinates, are like gigantic jellyfish with a single enormous eye. Scientifically, they are more advanced than the solar system, with far superior spaceships, weapons that toss miniature suns, and much else. * Despite their advancement, they are losing the battle for their planet. Their sun is dying, and the planetary environment is incredibly hostile. * Eric Ulnar has made a treaty with them. In exchange for a shipment of iron, which is lacking in their system, they will conquer the solar system for him and restore the Ulnar empire. Actually, the Medusae, who are expectedly vicious and foul, really mean to take the solar system for themselves. Eric learns this too late. * Personalities: The conspirators include Adam and Eric Ulnar; Adam is to some extent a man of honor, even if beset by schemes of power, while Eric is simply a wastrel and swine. * Against the conspirators are: John Ulnar, a distant relative, who in a wooden way is totally upright and loyal to Green Hall; Aladoree Anthar, a beautiful maiden, who is the current holder of [Earth's secret superweapon] AKKA; and the then equivalents of the Three Musketeers. They are Jay Kalam (mind and honor), Hal Samdu (muscle and bravery), and Giles Habibula (gluttony and deviousness, but honorably so). Giles is an ingenious rogue of many unexpected abilities. * The narrative: A gigantic Medusan ship, guided by Eric Ulnar, kidnaps Aladoree and takes here to [the alien homeworld] Yarkand, where the Medusans work on her to obtain the secret of AKKA. John Ulnar, at first deceived by the other Ulnars, finally recognizes what is happening and follows with his three companions to rescue her. After great peril they reach Yarkand, which is a horrible place, vividly imagined. They are captured, but fortunately are imprisoned near Aladoree, whom they rescue. Returning to the solar system, they find that the Medusae have all but conquered it, and that the human race is being destroyed by a poison gas (peculiar to Yarkand) that the Medusae are droping on Earth. Aladoree, who has suffered badly during her captivity and rescue, recovers sufficiently to set up an AKKA apparatus. She first disintegrates the Medusan fleet, then the Moon, which was the Medusan base. John Ulnar, whose name is now officially changed to Star, and Aladoree marry. * Miscellaneous: Space travel is accomplished by geodynes (which permit faster-than-light travel) and auxiliary rockets. Hand weapons include proton guns.