It is certainly true that northern whites generally did not want free or freed blacks entering their region. Such racial prejudice did not prevent the evolution of the USA policy toward emancipation and black enlistment.In terms of non-slave States, I do not think that on the whole they would have welcomed blacks coming to them as they were generally not wanted and were prohibited from joining the Union army (see  below].
Just as a point of information, thousands of enslaved people gained freedom before the final Emancipation Proclamation (EP) and black enlistment, viz: The thousands of enslaved people who used the war to seek and gain freedom was not an option at the beginning of the war. It was unlawful for black people to join the Union army at that time. The opportunity for black people to join the Union army did not arrive until 1863 when the war was about half over and even then the Union did not want them mixing with their other soldiers so they were segregated through the creation of the USCT to which you refer.
May 24: Fugitive slaves at Fortress Monroe, Virginia, are received and put to work by Union general Benjamin F. Butler, who declares them “contraband of war”
August 6: First Confiscation Act nullifies owners' claims to fugitive slaves who had been employed in the Confederate war effort
April 16 :The USA abolishes slavery in the District of Columbia, with compensation to loyal owners, and appropriates money for the voluntary removal (“colonization”) of former slaves to Haiti, Liberia, or other countries
July 17: Second Confiscation Act frees the slaves of persons engaged in or assisting the rebellion and provides for the seizure and sale of other property owned by disloyal citizens; it also forbids army and navy personnel to decide on the validity of any fugitive slave's claim to freedom or to surrender any fugitive to any claimant, and authorizes the president to employ “persons of African descent” in any capacity to suppress the rebellion
July 17 Militia Act provides for the employment of “persons of African descent” in “any military or naval service for which they may be found competent,” granting freedom to slaves so employed (and to their families if they belong to disloyal owners)
August 22: In New Orleans, General Benjamin F. Butler incorporates into Union forces several “Native Guard” units composed of free-black soldiers; soon thereafter he begins recruiting both free-black and ex-slave men for additional regiments
August 25: After having withheld its permission for months, the War Department authorizes recruitment of black soldiers in the South Carolina Sea Islands
As noted, fugitive slaves ran to Union lines as early as May 1861, and many did gain refuge. Thousands were freed based on the above measures before the EP. Note, for example, that in August of 1862 a group of concerned citizens in Liberty County, GA wrote the following to Confederate military officials: "Independent of the forcible seizure of slaves by the enemy whenever it lies in his power, and to which we now make no allusion... we may set down as a low estimate the number of slaves absconded and enticed off from our sea-board at 20,000, and their value at from $12,000,000 to $15,000,000..."
The 20,000 enslaved people who "absconded" (that is, ran away and gained freedom) through August 1862 consisted only of persons on the Confederate sea-board. Other enslaved people gained freedom further inland.
These slave escapes helped to evolve USA policy toward emancipation and black enlistment. Some 178,000 black men joined the army starting basically in 1863. The majority of the black enlistees were people of color from the Confederate states, mostly former enslaved people, but certainly including free blacks. Racial animus did not prevent the USA policy of emancipation and black enlistment. The evolution in USA policy continued after the war with the 14th and 15th Amendments.