Barry is right that these are irregular verb forms that have to be memorized and can’t be derived according to the normal rules for present stem and participle formation. However, I think Hugh is sufficiently sophisticated that a detailed explanation of how these forms arose is warranted. It may not be easy to follow, and there’s no need to remember all the details, but it will give some idea of the complexity of Greek verbs, and also how there are certain underlying regularities beneath the apparently wild irregularities.
The basis for ὑπισχνέομαι is the verbal prefix ὑπο + the verbal root *σεχ-, which has a “zero grade” form σχ-. (The zero grade form is represented by the second aorist εσχον).
At a point in the evolution of the Greek language before literacy, initial σ- was weakened. Usually it was weakened to a rough breathing, but in the case of the verb *σεχω and its derivatives, initial σ- was simply lost and the verb is represented orthographically with a smooth breathing, resulting in the familiar verb ἔχω. (The zero grade form is represented by the second aorist ἔ-σχ-ον) The loss of σ- without initial aspiration (rough breathing) was due to a rule of Greek phonology that prevented two successive aspirated sounds – an initial aspiration in place of σ- could not coexist with the -χ-.
However, the finite present indicative ὑπισχνέομαι was formed from an alternative present stem.
Some verbs form their present stem by reduplication: adding a syllable at the beginning with the initial consonant plus the vowel ι. (Perfect stems are typically formed by reduplication, too, but the vowel is ε.) So an alternative present stem was formed from the root σ(ε)χ-. An original reduplicated present stem would have been *σισχ-, but this became *ισχ- when initial σ- ceased to be pronounced. Again, no initial rough breathing due to the presence of -χ-.
In the Attic dialect, the finite present stem ὑπισχνέομαι was formed by adding both the verbal prefix ὑπο and a suffix, -νε-, with middle endings, to the reduplicated root in the form *-ισχ-. (The suffix -ν- was added to other verbal roots in various forms to form present stems; see Smyth section 523).
But the suffix, -νε- was not added to the present stem in the Ionic dialect and the language of Homeric epic, which have ὑπίσχομαι.
The middle participle ὑποσχόμενος was formed by adding the verbal prefix ὑπο to the zero-grade root -σχ- without reduplication, with the middle participle ending.
I don’t think this will help you remember the forms, and there’s no need to remember the details of these processes, but I hope it will give you an idea of how the Greek verb system evolved and produced so many seemingly “irregular” forms, and also a hint of the underlying regularities – without confusing you too much.