The high court’s appeal was further sought to be appealed to the UKSC but not first to the EWCA. Why was the Court of Appeal passed over in this case?
The decision which Assange sought to appeal, United States v Assange  EWHC 3313 (Admin), was a decision of a divisional court of three judges. That is because s 26 of the Extradition Act 2003 provides for an appeal from "the appropriate judge" to the High Court, not the Court of Appeal. A divisional court of the High Court, called the Administrative Court, exercises this appellate jurisdiction and as an intermediate appellate court of three High Court judges, has a similar status to the Court of Appeal.
When a point of law has already been considered by a single judge, and a panel of three judges on appeal, it makes sense for the Supreme Court to consider whether to grant leave to appeal directly under the provisions cited by Jen rather than requiring the parties to go through another intermediate court. An example of this is R (on the application of Highbury Poultry Farm Produce Ltd) v Crown Prosecution Service  UKSC 39, an appeal from a divisional court exercising judicial review jurisdiction. However, the Supreme Court refused leave to appeal from the divisional court in Assange's case.
I'm not sure the other answers explicitly address Assange's approach.
Assange sought permission to appeal on the basis that the High Court decision in December 2021 involved three questions of law of general public importance. Such a question is one that has a wider application than the specific case in which it is raised.
The three points of law were:
"Is the provision of assurances governed by the principles in Hungary v Fenyvesi  4 All ER 324?"
"Is it permissible to approach oppression under section 91 of the 2003 Act on the basis that it may be imposed if brought about by the defendant’s own alleged conduct?"
"Is it permissible to approach article 3 ECHR on the basis that inhuman or degrading treatment may be imposed if brought about by the defendant’s own alleged conduct?"
In January 2022 the High Court certified the first point as a question of law of general public importance. The court found that while the first point was "long settled" it had not been considered by the Supreme Court. The High Court asked the Supreme Court to expedite its consideration of Assange's application to it were he to make one. Normally the Supreme Court may take six months to a year to hear cases (unless urgent).
Assange did apply to the Supreme Court and in March 2022 the court ruled that Assange had not raised "an arguable point of law."